PGTRB UNIT 1: ALGEBRA - COMPLETE CONCEPTS & PRACTICE QUESTIONS

 

PGTRB UNIT 1: ALGEBRA - COMPLETE CONCEPTS

Groups:

• Binary Operator: Let $G$ be a non-empty set. A binary operation $\ast$ on $G$ satisfies the closure axiom: $\mathrm{\forall }a,b\in G\Rightarrow a\ast b\in G$. The structure $(G,\ast )$ is called a groupoid.

• Group: The structure $(G,\ast )$ is a group if it satisfies:

  1. Closure: $\mathrm{\forall }a,b\in G\Rightarrow a\ast b\in G$

  2. Associativity: $\mathrm{\forall }a,b,c\in G,a\ast (b\ast c)=(a\ast b)\ast c$

  3. Identity: $\mathrm{\exists }e\in G$ such that $a\ast e=e\ast a=a$

  4. Inverse: $\mathrm{\forall }a\in G,\mathrm{\exists }{a}^{-1}\in G$ such that $a\ast a^{-1}=a^{-1}\ast a=e$

• Abelian Group: A group $(G,\ast )$ is abelian if it is commutative: $\mathrm{\forall }a,b\in G\Rightarrow a\ast b=b\ast a$.

• Remarks:

  • Semigroup: Closure + Associativity.

  • Monoid: Closure + Associativity + Identity.

  • Trivial Group: $\{e\}$.

  • Finite/Infinite Group: Based on the number of elements.

• Order of a Group: Denoted by $O(G)$, it is the number of distinct elements in a finite group.

Properties of a Group:

1. The identity element is unique.

2. The inverse of an element is unique.

3. $e^{-1}=e$.

4. $(a^{-1}{)}^{-1}=a$.

5. $(a\ast b{)}^{-1}=b^{-1}\ast a^{-1}$.

6. Cancellation Laws: $a\ast b=a\ast c\Rightarrow b=c$ and $b\ast a=c\ast a\Rightarrow b=c$.

7. The equations $a\ast x=b$ and $y\ast a=b$ have unique solutions in $G$.

8. For an abelian group, $(a\cdot b{)}^n=a^n{b}^n$.

9. If $(a\cdot b{)}^2=a^2{b}^2$, then $G$ is abelian.

10. If $o(G)\le 4$, then $G$ is abelian.

Permutation Groups:

• A one-one mapping of a finite set $S$ onto itself is a permutation of degree $n$.

• Identity Permutation: $f(a)=a,\mathrm{\forall }a\in S$.

• Inverse Permutation: If $f(a)=b$, then $f^{-1}(b)=a$.

• Product/Composition: The product of two permutations $f$ and $g$ is denoted $fg$. In general, $fg\mathrm{\ne }gf$.

• Symmetric Group $S_n$: The group of all $n!$ permutations of a set of $n$ elements.

• Cyclic Permutation (Cycle): A permutation that cyclically permutes a subset of elements.

  • A cycle of length $m$ is an $m$-cycle.

  • A 2-cycle is a transposition.

  • A 1-cycle is the identity permutation.

• Remarks:

  • Every permutation can be expressed as a product of disjoint cycles.

  • Every permutation can be expressed as a product of transpositions.

  • A permutation is even if it can be expressed as a product of an even number of transpositions; otherwise, it is odd.

  • The identity permutation is even.

  • The product of two even (or odd) permutations is even.

  • The inverse of an even (or odd) permutation is even (or odd).

  • Every transposition is odd.

  • $S_n$ has $\frac{n!}{2}$ even and $\frac{n!}{2}$ odd permutations.

  • Alternating Group $A_n$: The set of all even permutations of $S_n$, which is a normal subgroup of index 2.

Cyclic Group:

• A group $G$ is cyclic if $\mathrm{\exists }a\in G$ such that every element $x\in G$ is of the form $a^n$ for some integer $n$. The element $a$ is called a generator of $G$.

• Remarks:

  1. Every cyclic group is abelian, but the converse is not true.

  2. If $a$ is a generator, then $a^{-1}$ is also a generator.

  3. $O(\text{group})=O(\text{generator})$.

  4. Every group of prime order is cyclic.

  5. Every subgroup of a cyclic group is cyclic.

  6. Every isomorphic image of a cyclic group is cyclic.

  7. Cyclic groups of the same order are isomorphic.

  8. If a finite group of order $n$ contains an element of order $n$, it is cyclic.

  9. The number of generators of a finite cyclic group of order $n$ is $\varphi (n)$ (Euler's totient function).

  10. An infinite cyclic group has precisely 2 generators.

  11. The converse of Lagrange's theorem holds for cyclic groups.

Lagrange's Theorem:

• The order of each subgroup of a finite group divides the order of the group.

• Index of $H$ in $G$: $\frac{O(G)}{O(H)}$.

• The converse of Lagrange's theorem is not true.

Cayley's Theorem:

• Every finite group is isomorphic to a permutation group (a subgroup of a symmetric group).

Isomorphism:

• A mapping $f:G\to G^{\mathrm{\prime }}$ is an isomorphism if it is a bijective homomorphism.

Subgroup:

• A non-empty subset $H$ of a group $(G,\ast )$ is a subgroup if $(H,\ast )$ is itself a group.

• The identity of a subgroup is the same as the group's identity.

• The intersection of two subgroups is a subgroup.

• The union of two subgroups is a subgroup iff one is contained in the other.

• If $H$ is a subgroup of $G$, then $H^{-1}=H$.

• If $H$ and $K$ are subgroups of an abelian group $G$, then $HK$ is a subgroup.

Euler's & Fermat's Theorems:

• Euler's: If $gcd(a,n)=1$, then $a^{\varphi (n)}\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}n$.

• Fermat's: If $p$ is prime, then $a^p\equiv a\mathrm{m}\mathrm{o}\mathrm{d}p$.

Normal Subgroups:

• A subgroup $H$ of $G$ is normal if $aH=Ha,\mathrm{\forall }a\in G$. Denoted $H◃G$.

• Simple Group: A group with no proper normal subgroups.

• The center $Z(G)$ of a group is a normal subgroup.

• A subgroup of index 2 is normal.

• Theorems:

  1. $H◃G$ iff $\mathrm{\forall }a\in G,h\in H,ah{a}^{-1}\in H$.

  2. The intersection of normal subgroups is normal.

  3. If $H◃G$ and $K$ is a subgroup of $G$ with $H\subseteq K\subseteq G$, then $H◃K$.

  4. If $N◃G$ and $H$ is any subgroup of $G$, then $NH$ is a subgroup of $G$ and $N◃NH$.

  5. If $N$ is a cyclic normal subgroup of $G$, then every subgroup of $N$ is normal in $G$.

Conjugate Element:

• $a$ is conjugate to $b$ if $\mathrm{\exists }x\in G$ such that $a=x^{-1}bx$.

Normalizer:

• $N(a)=\{x\in G:ax=xa\}$ is a subgroup of $G$.

Quotient Group:

• If $H◃G$, the set of cosets $G\mathrm{/}H=\{Ha:a\in G\}$ forms a group under coset multiplication.

• $O(G\mathrm{/}H)=\frac{O(G)}{O(H)}$.

• Every quotient group of a cyclic (or abelian) group is cyclic (or abelian).

Homomorphism:

• A mapping $f:G\to G^{\mathrm{\prime }}$ is a homomorphism if $f(xy)=f(x)f(y)$.

• Types: Isomorphism (bijective), Monomorphism (injective), Epimorphism (surjective).

• Endomorphism: Homomorphism of a group into itself.

• Automorphism: Isomorphism of a group onto itself.

• Kernel: $Ker(f)=\{x\in G:f(x)=e^{\mathrm{\prime }}\}$. The kernel is a normal subgroup of $G$.

• $f$ is an isomorphism iff $Ker(f)=\{e\}$.

• Fundamental Theorem: Every homomorphic image of $G$ is isomorphic to a quotient group of $G$ (i.e., $G\mathrm{/}Ker(f)\cong f(G)$).

Isomorphism Theorems:

1. First: Let $f:G\to G^{\mathrm{\prime }}$ be an epimorphism with kernel $K$. If $H^{\mathrm{\prime }}◃G^{\mathrm{\prime }}$ and $H=f^{-1}(H^{\mathrm{\prime }})$, then $G\mathrm{/}H\cong G^{\mathrm{\prime }}\mathrm{/}{H}^{\mathrm{\prime }}$.

2. Second: If $H$ and $N$ are normal subgroups of $G$ with $N\subseteq H$, then $G\mathrm{/}H\cong (G\mathrm{/}N)\mathrm{/}(H\mathrm{/}N)$.

3. Third: If $H$ is a subgroup and $N$ is a normal subgroup of $G$, then $H\mathrm{/}(H\cap N)\cong HN\mathrm{/}N$.

Cauchy's Theorem (for Abelian Groups):

• If a prime $p$ divides $O(G)$ for a finite abelian group $G$, then $G$ has an element of order $p$.

Sylow Theorems:

• Let $O(G)=p^{m}n$, where $p$ is prime and $p\nmid n$. A subgroup of order $p^m$ is a Sylow $p$-subgroup.

  1. First: Sylow $p$-subgroups exist.

  2. Second: Any two Sylow $p$-subgroups are conjugate.

  3. Third: The number of Sylow $p$-subgroups is of the form $1+kp$ and divides $O(G)$.

• If there is only one Sylow $p$-subgroup, it is normal.

Center of a Group:

• $Z(G)=\{a\in G:ax=xa\mathrm{\forall }x\in G\}$. It is a subgroup of $G$.

• Groups of order $p^2$ (p prime) are abelian.

Solvable and Nilpotent Groups:

• Solvable Group: $G^{(k)}=\{e\}$ for some $k$.

• Nilpotent Group: $Z_m(G)=G$ for some $m$.

• Every nilpotent group is solvable.

• A group of order $p^n$ is nilpotent.

• For $n>4$, $A_n$ is simple and $S_n$ is not solvable.

Rings:

• An algebraic structure $(R,+,\cdot )$ is a ring if:

  1. $(R,+)$ is an abelian group.

  2. $(R,\cdot )$ is closed and associative.

  3. Distributive laws hold.

• Commutative Ring: Multiplication is commutative.

• Ring with Unity: Contains a multiplicative identity (1).

• Null Ring / Zero Ring: $R=\{0\}$.

• Boolean Ring: $a^2=a,\mathrm{\forall }a\in R$.

• P-ring: $a^p=a,\mathrm{\forall }a\in R$.

• Idempotent Element: $a^2=a$.

• Zero Divisor: Non-zero elements $a,b$ such that $ab=0$.

• Cancellation Laws hold in a ring iff it has no zero divisors.

Integral Domain:

• A commutative ring with unity and no zero divisors.

• Examples: $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C},{\mathbb{Z}}_p$ (p prime).

Division Ring / Skew Field:

• A ring with unity where every non-zero element has a multiplicative inverse.

Field:

• A commutative division ring.

• Examples: $\mathbb{Q},\mathbb{R},\mathbb{C},{\mathbb{Z}}_p$.

• Every field is an integral domain. Every finite integral domain is a field.

Subrings:

• A non-empty subset $S$ of a ring $R$ is a subring if $(S,+,\cdot )$ is itself a ring.

• Test: $a,b\in S\Rightarrow a-b\in S$ and $a\cdot b\in S$.

• The intersection of subrings is a subring.

Subfield:

• A non-empty subset $K$ of a field $F$ is a subfield if $(K,+,\cdot )$ is a field.

• Test: $a,b\in K\Rightarrow a-b\in K$ and $a{b}^{-1}\in K$ (for $b\mathrm{\ne }0$).

Characteristic of a Ring:

• The smallest positive integer $n$ such that $na=0,\mathrm{\forall }a\in R$. If no such $n$ exists, the characteristic is 0.

• The characteristic of an integral domain (or field) is either 0 or a prime number.

• The order of a finite field is $p^n$.

Ideals:

• A non-empty subset $S$ of a ring $R$ is an ideal if:

  1. $(S,+)$ is a subgroup of $(R,+)$.

  2. $a\in S,r\in R\Rightarrow ra\in S$ and $ar\in S$.

• Left Ideal: $a\in S,r\in R\Rightarrow ra\in S$.

• Right Ideal: $a\in S,r\in R\Rightarrow ar\in S$.

• The intersection, sum, and product of two ideals are ideals.

Homomorphism of Rings:

• A mapping $f:R\to R^{\mathrm{\prime }}$ is a homomorphism if $f(a+b)=f(a)+f(b)$ and $f(ab)=f(a)f(b)$.

• $f(0)=0^{\mathrm{\prime }}$, $f(-a)=-f(a)$.

• $f(R)$ is a subring of $R^{\mathrm{\prime }}$.

• Kernel: $Ker(f)=\{a\in R:f(a)=0^{\mathrm{\prime }}\}$. The kernel is an ideal of $R$.

Isomorphism of Rings:

• A bijective homomorphism. The isomorphic image of an integral domain/field/division ring is an integral domain/field/division ring.

Quotient Rings:

• If $S$ is an ideal of $R$, then $R\mathrm{/}S=\{a+S:a\in R\}$ is a ring under coset operations.

• Fundamental Theorem: If $f:R\to R^{\mathrm{\prime }}$ is an epimorphism, then $R\mathrm{/}Ker(f)\cong R^{\mathrm{\prime }}$.

Principal Ideal:

• An ideal generated by a single element $a$, denoted $(a)=\{ar:r\in R\}$.

• Principal Ideal Domain (PID): An integral domain where every ideal is principal.

• Examples: $\mathbb{Z}$, any field.

Prime Ideals:

• An ideal $P\mathrm{\ne }R$ in a commutative ring $R$ is prime if $ab\in P\Rightarrow a\in P$ or $b\in P$.

• In a commutative ring with unity, every maximal ideal is prime.

Maximal Ideals:

• An ideal $M\mathrm{\ne }R$ is maximal if no ideal lies strictly between $M$ and $R$.

• In a commutative ring with unity, an ideal $M$ is maximal iff $R\mathrm{/}M$ is a field.

Quotient Field:

• Every integral domain can be embedded in a field (its field of fractions). The quotient field of $\mathbb{Z}$ is $\mathbb{Q}$.

Euclidean Domain:

• An integral domain $R$ with a function $d:R\setminus \{0\}\to {\mathbb{Z}}_{\ge 0}$ such that:

  1. $d(a)\le d(ab)$ for $a,b\mathrm{\ne }0$.

  2. $\mathrm{\forall }a,b\in R,b\mathrm{\ne }0,\mathrm{\exists }q,r\in R$ such that $a=bq+r$ with $r=0$ or $d(r)<d(b)$.

• Examples: $\mathbb{Z}$, any field, ring of Gaussian integers.

Unique Factorization Domain (UFD):

• An integral domain where every non-zero, non-unit element can be written as a unique product of irreducible elements (up to order and associates).

• Hierarchy: Euclidean Domain $\Rightarrow$ PID $\Rightarrow$ UFD.

Polynomial Rings:

• $R[x]$, the set of all polynomials over a ring $R$.

• Degree: The highest power of $x$ with a non-zero coefficient.

• If $R$ is an integral domain, then $R[x]$ is an integral domain and $deg(f(x)g(x))=degf(x)+degg(x)$.

• If $R$ is a UFD, then $R[x]$ is a UFD.

• If $F$ is a field, then $F[x]$ is a Euclidean Domain, a PID, and a UFD.

• Monic Polynomial: Leading coefficient is 1.

Irreducible Polynomial:

• A non-constant polynomial that cannot be factored into polynomials of lower degree.

• Examples: $x^2+1$ is irreducible over $\mathbb{R}$. $x^2-2$ is irreducible over $\mathbb{Q}$.

Vector Spaces:

• A set $V$ over a field $F$ with vector addition and scalar multiplication satisfying:

  1. $(V,+)$ is an abelian group.

  2. Closure and compatibility conditions for scalar multiplication.

• Examples: ${\mathbb{R}}^n$, $M_{m\times n}(F)$, $F[x]$, $C[a,b]$.

Subspaces:

• A non-empty subset $W$ of a vector space $V(F)$ is a subspace if it is itself a vector space.

• Test: $\alpha ,\beta \in W\Rightarrow \alpha -\beta \in W$; $a\in F,\alpha \in W\Rightarrow a\alpha \in W$.

• The intersection of subspaces is a subspace.

Linear Combination & Linear Span:

• Linear Combination: $\alpha =a_1{\alpha }_1+a_2{\alpha }_2+\cdots +a_n{\alpha }_n$.

• Linear Span $L(S)$: The set of all linear combinations of a subset $S$. It is the smallest subspace containing $S$.

Linear Dependence/Independence:

• A set $\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\}$ is linearly dependent if there exist scalars, not all zero, such that $a_1{\alpha }_1+\cdots +a_n{\alpha }_n=0$. Otherwise, it is linearly independent.

Basis and Dimension:

• Basis: A linearly independent set that spans the vector space.

• Dimension: The number of elements in a basis.

• Theorems:

  • Every finite-dimensional vector space has a basis.

  • All bases have the same size (dimension).

  • Any linearly independent set can be extended to a basis.

  • $dim(W_1+W_2)=dim{W}_1+dim{W}_2-dim(W_1\cap W_2)$.

Linear Transformations:

• A mapping $T:U\to V$ between vector spaces over $F$ is linear if $T(a\alpha +b\beta )=aT(\alpha )+bT(\beta )$.

• Kernel/Null Space: $Ker(T)=\{\alpha \in U:T(\alpha )=0\}$. It is a subspace of $U$.

• Range/Image: $R(T)=\{T(\alpha ):\alpha \in U\}$. It is a subspace of $V$.

• Rank-Nullity Theorem: $dim(U)=rank(T)+nullity(T)$.

Isomorphism of Vector Spaces:

• A bijective linear transformation. Two finite-dimensional vector spaces are isomorphic iff they have the same dimension.

Quotient Space:

• $V\mathrm{/}W=\{v+W:v\in V\}$, where $W$ is a subspace of $V$. $dim(V\mathrm{/}W)=dimV-dimW$.

Direct Sum:

• $V=W_1\oplus W_2$ if $V=W_1+W_2$ and $W_1\cap W_2=\{0\}$.

Linear Functional & Dual Space:

• Linear Functional: A linear transformation from $V$ to its field $F$.

• Dual Space $V^{\ast }$: The vector space of all linear functionals on $V$. $dim(V^{\ast })=dim(V)$.

• Dual Basis: If $B=\{{\alpha }_1,\dots ,{\alpha }_n\}$ is a basis for $V$, then the dual basis $B^{\ast }=\{f_1,\dots ,f_n\}$ for $V^{\ast }$ is defined by $f_i({\alpha }_j)={\delta }_{ij}$.

• Annihilator: For a subset $S\subseteq V$, $S^{\circ }=\{f\in V^{\ast }:f(\alpha )=0\mathrm{\forall }\alpha \in S\}$. $dimW+dim{W}^{\circ }=dimV$.

Minimal Polynomial:

• The monic polynomial of least degree that annihilates a linear operator $T$.

Cayley-Hamilton Theorem:

• Every linear operator on a finite-dimensional vector space satisfies its own characteristic equation.

Diagonalizable Operator:

• An operator is diagonalizable if there exists a basis of $V$ consisting of eigenvectors of $T$.

Finite Fields (Galois Fields):

• A field with a finite number of elements, $p^n$, where $p$ is prime.

• The multiplicative group of a finite field is cyclic.

• Every element satisfies $x^{p^n}=x$.

Field Extensions:

• A field $K$ is an extension of $F$ if $F\subseteq K$. Denoted $K\mathrm{/}F$.

• Degree of Extension $[K:F]$: The dimension of $K$ as a vector space over $F$.

• Finite Extension: $[K:F]$ is finite.

• Tower Law: If $F\subseteq K\subseteq L$, then $[L:F]=[L:K][K:F]$.

Algebraic Elements:

• An element $\alpha \in K$ is algebraic over $F$ if it is a root of some non-zero polynomial in $F[x]$.

• Minimal Polynomial: The monic irreducible polynomial in $F[x]$ of which $\alpha$ is a root. Its degree is the degree of $\alpha$.

• $F(\alpha )$ is a finite extension of $F$ with $[F(\alpha ):F]=\text{degree of the minimal polynomial}$.

Algebraic Extension:

• An extension where every element of $K$ is algebraic over $F$. Every finite extension is algebraic.

Splitting Field:

• The smallest field extension $K\mathrm{/}F$ over which a polynomial $f(x)$ factors into linear factors.

Separable Extension:

• An extension where the minimal polynomial of every element has no repeated roots.

• Perfect Field: A field over which all finite extensions are separable (e.g., fields of characteristic 0).

Galois Theory:

• Galois Group $G(K\mathrm{/}F)$: The group of all $F$-automorphisms of $K$.

• Fixed Field: The set of elements in $K$ fixed by every automorphism in a subgroup of $G(K\mathrm{/}F)$.

• Galois Extension: A finite, normal, and separable extension.

• Fundamental Theorem of Galois Theory: Establishes a one-to-one correspondence between the subfields of a Galois extension $K\mathrm{/}F$ and the subgroups of $G(K\mathrm{/}F)$.

Inner Product Spaces:

• A vector space with an inner product $⟨\cdot ,\cdot ⟩$.

• Norm: $\mathrm{\parallel }\alpha \mathrm{\parallel }=\sqrt{⟨\alpha ,\alpha ⟩}$.

• Schwarz's Inequality: $\mathrm{\mid }⟨\alpha ,\beta ⟩\mathrm{\mid }\le \mathrm{\parallel }\alpha \mathrm{\parallel }\mathrm{\parallel }\beta \mathrm{\parallel }$.

• Triangle Inequality: $\mathrm{\parallel }\alpha +\beta \mathrm{\parallel }\le \mathrm{\parallel }\alpha \mathrm{\parallel }+\mathrm{\parallel }\beta \mathrm{\parallel }$.

• Orthogonal Vectors: $⟨\alpha ,\beta ⟩=0$.

• Orthonormal Set: An orthogonal set where each vector has norm 1.

• Gram-Schmidt Orthogonalization: A process for converting a basis into an orthogonal/orthonormal basis.
  
  

  Complete List of Problems & Examples from the PDF

  1. Group Examples & Model Problems

  • Example 1: The set $\mathbb{R}$ of all real numbers forms an abelian group under $a\ast b=a+b+1$.

  • Example 2: The sets $\{1,\omega ,{\omega }^2\}$ (cube roots of unity) and $\{1,-1,i,-i\}$ (fourth roots of unity) form an abelian group under multiplication.

  • Example 3: The set $G=\{a\in \mathbb{R}\mid a\mathrm{\ne }1\}$ forms an abelian group under $a\ast b=a+b-ab$.

  • Example 4: The set $G=\{a\in \mathbb{R}\mid a\mathrm{\ne }-1\}$ forms a group under $a\ast b=a+b+ab$.

  • Example 5: The set of matrices $\left(\begin{array}{cc}a& b\\ -b& a\end{array}\right)$ where $a,b\in \mathbb{R}$ and $a^2+b^2\mathrm{\ne }0$, forms an infinite abelian group under matrix multiplication.

  • Example 6: $({\mathbb{Z}}_n,{+}_n)$ is a group. $({\mathbb{Z}}_n\setminus \{0\},{\times }_n)$ is an abelian group if $n$ is prime.

  • Example 7: The set of all bilinear transformations $f(z)=\frac{az+b}{cz+d}$, where $ad-bc\mathrm{\ne }0$, $a,b,c,d\in \mathbb{C}$, is an infinite abelian group.

  • Example 8: $(\mathbb{N},+)$ is a semigroup. $(\mathbb{N},\ast )$ is a monoid.

  • Example 9: $(\mathbb{Z},+)$ is an abelian group. $(\mathbb{Z},\ast )$ is a monoid.

  • Example 10: $(\mathbb{R},+)$ and $(\mathbb{C},+)$ are abelian groups. $(\mathbb{R}\setminus \{0\},\ast )$ and $(\mathbb{C}\setminus \{0\},\ast )$ are abelian groups.

  2. Permutation & Symmetric Group Problems

  • Example: For $S=\{1,2,3,4\}$, given permutations:
    $f=\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 4& 1& 3\end{array}\right)$, $g=\left(\begin{array}{cccc}1& 2& 3& 4\\ 1& 3& 2& 4\end{array}\right)$.

    • Find the product $fg=\left(\begin{array}{cccc}1& 2& 3& 4\\ 4& 2& 1& 3\end{array}\right)$.

    • Find the product $gf=\left(\begin{array}{cccc}1& 2& 3& 4\\ 4& 1& 3& 2\end{array}\right)$.

  • Example: Show that $(123)$ is a cycle of length 3.

  • Example: Show that $(12345)$ is a cycle of length 5.

  • Example: Show that $(12)(34)$ is not a cycle.

  • Example: Find the product of cycles $(123)(5641)=(123564)$.

  • Problem: Find the number of distinct 3-cycles in $S_4$.

    • Solution: $\frac{1}{3}\cdot \frac{4!}{(4-3)!}=\frac{1}{3}\cdot \frac{24}{1}=8$.

  3. Cyclic Group Problems

  • Example 1: $G=\{1,-1,i,-i\}$ under multiplication is cyclic. Generators: $i,-i$.

  • Example 2: $G=\{1,\omega ,{\omega }^2\}$ under multiplication is cyclic. Generators: $\omega ,{\omega }^2$.

  • Example 3: $G=\{0,1,2,3,4,5\}$ under addition modulo 6 is cyclic. Generators: $1,5$.

  • Example 4: The set of $n^{th}$ roots of unity is cyclic. Generator: $e^{\frac{2\pi i}{n}}$.

  • Problem: Find the number of generators of a finite cyclic group of order 28.

    • Solution: $\varphi (28)=28\times (1-\frac{1}{2})\times (1-\frac{1}{7})=28\times \frac{1}{2}\times \frac{6}{7}=12$.

  4. Sylow Theorem & Group Structure Problems

  • Example 1: Show that a group of order 45 is abelian.

  • Example 2: Show that there is no simple group of order 120.

  • Problem: The number of elements of order $d$ in a cyclic group of order $n$ is $\varphi (d)$.

  5. Ring and Field Examples

  • Examples of Rings: $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C},M_n(\mathbb{R})$ (non-commutative), ${\mathbb{Z}}_n$.

  • Example of Boolean Ring: ${\mathbb{Z}}_2=\{0,1\}$.

  • Example of Ring without unit: The set of all even integers.

  • Examples of Rings with zero divisors: The ring of all $2\times 2$ matrices over integers; $({\mathbb{Z}}_6,{+}_6,{\times }_6)$.

  • Examples of Rings without zero divisors: $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$.

  • Examples of Integral Domains: $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C},{\mathbb{Z}}_p$ (p prime).

  • Examples of Fields: $\mathbb{Q},\mathbb{R},\mathbb{C},{\mathbb{Z}}_p$ (p prime).

  • Example: The set of numbers of the form $a+b\sqrt{2}$, $a,b\in \mathbb{Q}$, is a field.

  • Example: The Gaussian integers $\{a+ib\mid a,b\in \mathbb{Z}\}$ form a commutative ring with unity and an integral domain, but not a field.

  • Example: The set $\{b\sqrt{2}\mid b\in \mathbb{Q}\}$ is not a ring.

  6. Ideal Examples

  • Example: $\left\{\left(\begin{array}{cc}a& 0\\ b& 0\end{array}\right)\right\}$ is a left ideal of the ring of all $2\times 2$ matrices.

  • Example: $(2)$ and $(3)$ are ideals in $\mathbb{Z}$.

  7. Polynomial Irreducibility Problems

  • Problem 1:

  • Show $x^2+1$ is irreducible over $\mathbb{R}$.

    Explanation:
    Over $\mathbb{R}$, irreducibility means it cannot be factored into polynomials of lower degree with real coefficients.
    The roots of $x^2+1=0$ are $x=\pm i$, which are not real.
    A quadratic with no real roots is irreducible over $\mathbb{R}$.
    So $x^2+1$ is irreducible over $\mathbb{R}$.

    ---

    Problem 2:

    Show $x^2-2$ is irreducible over $\mathbb{Q}$.

    Explanation:
    By Eisenstein’s criterion at $p=2$:
    $x^2-2$ has coefficients $1,0,-2$.
    $2$ divides $-2$ but $2^2=4$ does not divide $-2$, and $2$ does not divide the leading coefficient $1$.
    So Eisenstein applies ⇒ irreducible over $\mathbb{Q}$.

    ---

    Problem 3:

    Show $1+x+x^2+\cdots +x^{p-1}$ is irreducible over $\mathbb{Q}$ if $p$ is prime.

    Explanation:
    This is the $p$-th cyclotomic polynomial for prime $p$:

    $$1+x+\cdots +x^{p-1}=\frac{x^p-1}{x-1}\mathrm{.}$$

    It is known to be irreducible over $\mathbb{Q}$ (by Eisenstein after substitution $x=y+1$:

    $$\frac{(y+1{)}^p-1}{y}=y^{p-1}+\left(\genfrac{}{}{0px}{}{p}{1}\right)y^{p-2}+\cdots +\left(\genfrac{}{}{0px}{}{p}{p-1}\right)$$

    and $p$ divides each binomial coefficient $\left(\genfrac{}{}{0px}{}{p}{k}\right)$ for $1\le k\le p-1$, and $p^2$ does not divide the constant term $\left(\genfrac{}{}{0px}{}{p}{p-1}\right)=p$.
    So Eisenstein ⇒ irreducible.

    ---

    Problem 4:

    Show $x^2+1$ is irreducible over ${\mathbb{Z}}_7$.

    Explanation:
    Check if it has roots in ${\mathbb{Z}}_7$:
    $x^2\equiv -1\equiv 6(\mathrm{m}\mathrm{o}\mathrm{d}7)$.
    Squares mod 7: $0^2=0,1^2=1,2^2=4,3^2=2,4^2=2,5^2=4,6^2=1$.
    None equal 6 ⇒ no roots ⇒ irreducible over ${\mathbb{Z}}_7$.

    ---

    Problem 5:

    Show $x^2+x+5$ is reducible over ${\mathbb{Z}}_{11}$ since $x^2+x+5=(x+3)(x+9)$ in ${\mathbb{Z}}_{11}$.

    Explanation:
    Multiply: $(x+3)(x+9)=x^2+(3+9)x+27$.
    In ${\mathbb{Z}}_{11}$: $3+9=12\equiv 1$, $27\equiv 5$.
    So indeed $x^2+x+5$.
    Thus reducible.

    ---

    Problem 6:

    Show $x^3-9$ is reducible over ${\mathbb{Z}}_{11}$ since $x^3-9=(x+7)(x^2+4x+5)$.

    Explanation:
    Check: $x=-7\equiv 4$ in mod 11? Wait, $-7\equiv 4$ mod 11.
    Test $x=4$: $4^3-9=64-9=55\equiv 0$ mod 11 ⇒ yes, root.
    So factor out $(x-4)\equiv (x+7)$.
    Polynomial division yields $x^2+4x+5$.
    Thus reducible.

    ---

    Problem 7:

    Show $x^2+x+4$ is irreducible over ${\mathbb{Z}}_{11}$.

    Explanation:
    Discriminant: $1-16=-15\equiv -4\equiv 7$ mod 11.
    Check if 7 is a square mod 11: squares mod 11: $0,1,4,9,5,3$.
    7 is not in the list ⇒ no roots ⇒ irreducible.

  8. Vector Space & Basis Problems

  • Problem 1: Determine if $S=\{(1,2,4),(1,0,0),(0,1,0),(0,0,1)\}$ is linearly dependent. (Answer: Yes)

  • Problem 2: Determine if $S=\{(2,1,2),(8,4,8)\}$ is linearly dependent. (Answer: Yes)

  • Problem 3: Determine if $S=\{(1,2,0),(0,3,1),(-1,0,1)\}$ is linearly independent. (Answer: Yes)

  • Problem 4: Show that $\{1,x,x(1-x)\}$ is linearly independent.

  • Problem 5: Show that $(1,2,1),(2,1,0),(1,-1,2)$ form a basis of ${\mathbb{R}}^3$.

  • 8. Vector Space & Basis Problems

    Problem 1:

    $S=\{(1,2,4),(1,0,0),(0,1,0),(0,0,1)\}$ — linearly dependent?
    Answer: Yes — 4 vectors in ${\mathbb{R}}^3$ ⇒ must be dependent.

    ---

    Problem 2:

    $S=\{(2,1,2),(8,4,8)\}$ — linearly dependent?
    Answer: Yes — second vector = 4 × first ⇒ dependent.

    ---

    Problem 3:

    $S=\{(1,2,0),(0,3,1),(-1,0,1)\}$ — linearly independent?
    Answer: Yes — matrix formed by these rows has determinant $1(3\cdot 1-1\cdot 0)-2(0\cdot 1-1\cdot (-1))+0=3-2(1)=1\mathrm{\ne }0$.

    ---

    Problem 4:

    Show $\{1,x,x(1-x)\}$ is linearly independent.

    Explanation:
    Suppose $a\cdot 1+b\cdot x+c\cdot (x-x^2)=0$ as polynomials.
    That is $a+bx+cx-c{x}^2=a+(b+c)x-c{x}^2=0$.
    Then $a=0,b+c=0,-c=0$ ⇒ $c=0,b=0,a=0$.
    So independent.

    ---

    Problem 5:

    Show $(1,2,1),(2,1,0),(1,-1,2)$ form a basis of ${\mathbb{R}}^3$.

    Explanation:
    Check determinant of

    $$\left(\begin{array}{ccc}1& 2& 1\\ 2& 1& -1\\ 1& 0& 2\end{array}\right)$$

    = $1(1\cdot 2-(-1)\cdot 0)-2(2\cdot 2-(-1)\cdot 1)+1(2\cdot 0-1\cdot 1)$
    = $1(2)-2(4+1)+(0-1)$
    = $2-10-1=-9\mathrm{\ne }0$.
    So linearly independent, 3 vectors in ${\mathbb{R}}^3$ ⇒ basis.

  9. Minimal Polynomial Problems

  • • • $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$.

        • Solution: $A^2=3A$, so the minimal polynomial is $x^2-3x=0$.

    • Problem 1:

      $A=\left(\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)$.
      Find minimal polynomial. 

      Explanation:
      $A^2=3A$ (since each entry = 3).
      So $A^2-3A=0$ ⇒ minimal polynomial divides $x^2-3x$.
      $A\mathrm{\ne }0$ ⇒ not $x$.
      So minimal polynomial = $x^2-3x$.

      ---

      Problem 2:

      $A=\left(\begin{array}{cc}7& 4\\ 4& 7\end{array}\right)$.
      Minimal polynomial?

      Explanation:
      Char poly: $(7-\lambda {)}^2-16={\lambda }^2-14\lambda +33$.
      Distinct roots ⇒ minimal = characteristic = ${\lambda }^2-14\lambda +33$.

    • • Problem 2: Find the minimal polynomial of 

    • • $A=\left[\begin{array}{cc}7& 4\\ 4& 7\end{array}\right]$.

        • Solution: The characteristic polynomial is ${\lambda }^2-14\lambda +33=0$, which is also the minimal polynomi

  10. Field Extension & Splitting Field Problems

  • Problem 1: Show that $\sqrt{2}$ is algebraic over $\mathbb{Q}$ of degree 2. (Minimal polynomial: $x^2-2$)

  • Problem 2: Show that $\sqrt{2}+\sqrt{3}$ is algebraic over $\mathbb{Q}$. (Minimal polynomial: $x^4-10x^2+1$)

  • Problem 3: Show that $\sqrt[3]{2}$ is algebraic over $\mathbb{Q}$ of degree 3. (Minimal polynomial: $x^3-2$)

  • Problem 4: Find the splitting field and its degree over $\mathbb{Q}$ for $f(x)=x^4-5x^2+6$.

    • Solution: $f(x)=(x^2-3)(x^2-2)$. Splitting field: $\mathbb{Q}(\sqrt{2},\sqrt{3})$. Degree: $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}]=4$.

  • Problem 5: Find the splitting field of $x^4+1$ over $\mathbb{Q}$.

    • Solution: Splitting field: $\mathbb{Q}(\sqrt{2},i)$. Degree: 4.

  • Problem 6: Find the degree of the splitting field of $x^3-2$ over $\mathbb{Q}$. (Answer: 6)

  • Problem 7: Find the splitting field of $x^4+x^2+1$ over $\mathbb{Q}$.

    • Solution: $f(x)=(x^2+x+1)(x^2-x+1)$. The roots are primitive 3rd and 6th roots of unity. Splitting field: $\mathbb{Q}(\omega )$ where $\omega =\frac{-1+i\sqrt{3}}{2}$. Degree: 2.

  • Problem 8: Find the splitting field and degree of $x^4-2$ over $\mathbb{Q}$.

    • Solution: Splitting field: $\mathbb{Q}(\sqrt[4]{2},i)$. Degree: 8.

      Problem 1:

      $\sqrt{2}$ algebraic over $\mathbb{Q}$ of degree 2.
      Minimal polynomial: $x^2-2$.

      ---

      Problem 2:

      $\sqrt{2}+\sqrt{3}$ algebraic over $\mathbb{Q}$.
      Minimal polynomial: $x^4-10x^2+1$.

      ---

      Problem 3:

      $\sqrt[3]{2}$ algebraic over $\mathbb{Q}$ of degree 3.
      Minimal polynomial: $x^3-2$.

      ---

      Problem 4:

      $f(x)=x^4-5x^2+6=(x^2-3)(x^2-2)$.
      Splitting field: $\mathbb{Q}(\sqrt{2},\sqrt{3})$.
      Degree: $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}]=4$.

      ---

      Problem 5:

      Splitting field of $x^4+1$ over $\mathbb{Q}$.
      Roots: $e^{\pi i\mathrm{/}4},e^{3\pi i\mathrm{/}4},e^{5\pi i\mathrm{/}4},e^{7\pi i\mathrm{/}4}$ ⇒ $\mathbb{Q}(\sqrt{2},i)$.
      Degree: 4.

      ---

      Problem 6:

      Degree of splitting field of $x^3-2$ over $\mathbb{Q}$:
      Splitting field: $\mathbb{Q}(\sqrt[3]{2},\omega )$, ${\omega }^2+\omega +1=0$.
      Degree: $6$.

      ---

      Problem 7:

      $x^4+x^2+1=(x^2+x+1)(x^2-x+1)$.
      Roots: primitive 6th and 3rd roots of unity ⇒ splitting field $\mathbb{Q}({\omega }_3)=\mathbb{Q}(\sqrt{-3})$.
      Degree: 2.

      ---

      Problem 8:

      Splitting field of $x^4-2$ over $\mathbb{Q}$:
      $\mathbb{Q}(\sqrt[4]{2},i)$, degree 8.

  11. Inner Product Space & Gram-Schmidt Problem

  • Problem: Apply the Gram-Schmidt process to the basis $B=\{(1,0,1),(1,0,-1),(0,5,4)\}$ of ${\mathbb{R}}^3$ to find an orthonormal basis.

    • Solution Steps:

      1. ${\alpha }_1=\frac{P_1}{\mathrm{\parallel }{P}_1\mathrm{\parallel }}=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$

      2. $v_2=P_2-(P_2\cdot {\alpha }_1){\alpha }_1=(1,0,-1)$. ${\alpha }_2=\frac{v_2}{\mathrm{\parallel }{v}_2\mathrm{\parallel }}=(\frac{1}{\sqrt{2}},0,\frac{-1}{\sqrt{2}})$

      3. $v_3=P_3-(P_3\cdot {\alpha }_1){\alpha }_1-(P_3\cdot {\alpha }_2){\alpha }_2=(0,5,0)$. ${\alpha }_3=\frac{v_3}{\mathrm{\parallel }{v}_3\mathrm{\parallel }}=(0,1,0)$

    • Orthonormal Basis: $\{(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}),(\frac{1}{\sqrt{2}},0,\frac{-1}{\sqrt{2}}),(0,1,0)\}$

• 11. Gram–Schmidt Process

  Basis $B=\{(1,0,1),(1,0,-1),(0,5,4)\}$.

  Let $u_1=(1,0,1)$.
  $\mathrm{\parallel }{u}_1\mathrm{\parallel }=\sqrt{2}$ ⇒ $e_1=\frac{1}{\sqrt{2}}(1,0,1)$.

  $u_2=(1,0,-1)-[(1,0,-1)\cdot e_1]e_1$.
  $(1,0,-1)\cdot e_1=\frac{1}{\sqrt{2}}(1+0-1)=0$ ⇒ $u_2=(1,0,-1)$.
  $\mathrm{\parallel }{u}_2\mathrm{\parallel }=\sqrt{2}$ ⇒ $e_2=\frac{1}{\sqrt{2}}(1,0,-1)$.

  $u_3=(0,5,4)-[(0,5,4)\cdot e_1]e_1-[(0,5,4)\cdot e_2]e_2$.
  $(0,5,4)\cdot e_1=\frac{1}{\sqrt{2}}(0+0+4)=\frac{4}{\sqrt{2}}=2\sqrt{2}$.
  $(0,5,4)\cdot e_2=\frac{1}{\sqrt{2}}(0+0-4)=-2\sqrt{2}$.

  So $u_3=(0,5,4)-(2\sqrt{2})e_1-(-2\sqrt{2})e_2$
  = $(0,5,4)-(2,0,2)-(-2,0,2)$? Wait carefully:

  $(2\sqrt{2})e_1=(2\sqrt{2})\cdot \frac{1}{\sqrt{2}}(1,0,1)=(2,0,2)$.
  $(-2\sqrt{2})e_2=(-2\sqrt{2})\cdot \frac{1}{\sqrt{2}}(1,0,-1)=(-2,0,2)$.

  So $u_3=(0,5,4)-(2,0,2)-(-2,0,2)=(0,5,4)-(0,0,4)=(0,5,0)$.

  $\mathrm{\parallel }{u}_3\mathrm{\parallel }=5$ ⇒ $e_3=(0,1,0)$.

  Orthonormal basis:

  $$\{(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}),(\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}}),(0,1,0)\}\mathrm{.}$$

Group Theory

1. Define a Group.

• A group (G, *) is a set G with a binary operation * that satisfies: Closure, Associativity, Identity element (e), and Inverse element for every a in G.

2. What is an Abelian group?

• A group where the operation is commutative: a * b = b * a for all a, b in G.

3. Differentiate between a semi-group and a monoid.

• A semi-group satisfies closure and associativity. A monoid is a semi-group that also has an identity element.

4. What is the order of a group?

• The number of distinct elements in a finite group, denoted by O(G).

5. State two key properties of a group.

• The identity element is unique. The inverse of every element is unique.

6. If (a * b)² = a² * b² for all a, b in a group G, what can be concluded?

• The group G is Abelian.

7. What is the necessary and sufficient condition for a non-empty subset H of a group G to be a subgroup?

• For all a, b in H, a * b⁻¹ must also be in H.

8. Define the order of an element 'a' in a group G.

• The smallest positive integer n such that aⁿ = e (the identity element). If no such n exists, the element has infinite order.

9. State Lagrange's Theorem.

• For a finite group G and a subgroup H of G, the order of H divides the order of G.

10. Does the converse of Lagrange's Theorem hold?
- No, not in general. If a number d divides O(G), there may not be a subgroup of order d.

11. What is a cyclic group?
- A group that can be generated by a single element. Every element is of the form aⁿ for some integer n.

12. Is every cyclic group Abelian?
- Yes.

13. Is every Abelian group cyclic?
- No. (e.g., The Klein four-group V₄ is Abelian but not cyclic).

14. How many generators does a finite cyclic group of order n have?
- φ(n), where φ is Euler's totient function.

15. What is a permutation group?
- A group whose elements are permutations of a finite set and whose operation is composition of permutations.

16. What is the symmetric group Sₙ?
- The group of all permutations of a set of n elements. Its order is n!.

17. What is a transposition?
- A cycle of length 2; a permutation that swaps two elements and fixes the rest.

18. Define an even permutation.
- A permutation that can be expressed as a product of an even number of transpositions.

19. What is the alternating group Aₙ?
- The subgroup of Sₙ consisting of all even permutations. Its order is n!/2.

20. What is a normal subgroup?
- A subgroup H of G is normal if for every a in G, aH = Ha. Denoted by H ◁ G.

21. State a condition for a subgroup H to be normal.
- H is normal if for every a in G and h in H, a h a⁻¹ is in H.

22. What is a simple group?
- A group that has no non-trivial proper normal subgroups.

23. Define a coset.
- For a subgroup H of G and an element a in G, the left coset is aH = {ah | h in H}. The right coset is Ha = {ha | h in H}.

24. What is the index of a subgroup H in a group G?
- The number of distinct left (or right) cosets of H in G. It is equal to O(G)/O(H).

25. What is a quotient group (or factor group)?
- If H is a normal subgroup of G, the set of all cosets of H in G, denoted G/H, forms a group under coset multiplication.

26. Define a group homomorphism.
- A map f: G → G' between two groups such that f(ab) = f(a)f(b) for all a, b in G.

27. What is the kernel of a homomorphism?
- The set of elements in G that map to the identity in G'. Ker(f) = {a in G | f(a) = e'}.

28. What is an isomorphism?
- A homomorphism that is both one-to-one (injective) and onto (surjective).

29. State the Fundamental Theorem of Homomorphism for groups.
- If f: G → G' is a surjective homomorphism with kernel K, then G/K ≅ G'.

30. What does Cayley's theorem state?
- Every finite group is isomorphic to a subgroup of a symmetric group.

31. State Cauchy's Theorem for finite Abelian groups.
- If a prime p divides the order of a finite Abelian group G, then G has an element of order p.

32. What is a Sylow p-subgroup?
- For a group G of order pᵐ * n, where p does not divide n, a Sylow p-subgroup is a subgroup of order pᵐ.

33. State Sylow's First Theorem.
- For every prime power pᵏ dividing |G|, there exists a subgroup of order pᵏ.

34. State Sylow's Second Theorem.
- All Sylow p-subgroups of a group G are conjugate to each other.

35. State Sylow's Third Theorem.
- The number of Sylow p-subgroups, n_p, satisfies n_p ≡ 1 (mod p) and n_p divides |G|.

36. If a group has only one Sylow p-subgroup, what is true about that subgroup?
- It is a normal subgroup.

37. What is the center of a group Z(G)?
- Z(G) = {a in G | ag = ga for all g in G}. It is always a normal subgroup.

38. Is a group of order p² (p prime) always Abelian?
- Yes.

39. What is a solvable group?
- A group that has a subnormal series whose factor groups are all Abelian.

40. What is a nilpotent group?
- A group that has a central series terminating in G.

41. Is every nilpotent group solvable?
- Yes.

42. Is Sₙ solvable for n > 4?
- No, Sₙ is not solvable for n > 4.

---

Ring Theory

43. Define a Ring.
- A set R with two binary operations, + (addition) and ⋅ (multiplication), such that (R, +) is an Abelian group, multiplication is associative, and multiplication distributes over addition.

44. What is a commutative ring?
- A ring where multiplication is commutative: a ⋅ b = b ⋅ a for all a, b.

45. What is a ring with unity?
- A ring that has a multiplicative identity element, usually denoted by 1.

46. What is a zero divisor?
- A non-zero element a in a ring R for which there exists a non-zero element b such that a ⋅ b = 0.

47. Define an Integral Domain.
- A commutative ring with unity that has no zero divisors.

48. Give an example of an integral domain that is not a field.
- The ring of integers, Z.

49. What is a Field?
- A commutative ring with unity in which every non-zero element has a multiplicative inverse.

50. What is a Division Ring (or Skew Field)?
- A ring with unity where every non-zero element has a multiplicative inverse, but multiplication is not necessarily commutative.

51. Is every field an integral domain?
- Yes.

52. Is every integral domain a field?
- No (e.g., Z). But every finite integral domain is a field.

53. Define a subring.
- A subset S of a ring R that is itself a ring under the operations of R.

54. What is the characteristic of a ring?
- The smallest positive integer n such that n⋅a = 0 for all a in R. If no such n exists, the characteristic is 0.

55. What is the characteristic of an integral domain?
- Either 0 or a prime number.

56. What is an ideal of a ring?
- A subset I of a ring R such that (I, +) is a subgroup of (R, +) and for every r in R and a in I, both r⋅a and a⋅r are in I.

57. What is a principal ideal?
- An ideal generated by a single element a, denoted (a) = { r⋅a | r in R }.

58. What is a prime ideal?
- A proper ideal P in a commutative ring R such that if ab ∈ P, then either a ∈ P or b ∈ P.

59. What is a maximal ideal?
- A proper ideal M of R such that there is no proper ideal I of R with M ⊂ I ⊂ R.

60. If M is a maximal ideal of a commutative ring R with unity, what is R/M?
- R/M is a field.

61. If P is a prime ideal of a commutative ring R with unity, what is R/P?
- R/P is an integral domain.

62. Define a ring homomorphism.
- A map f: R → S between two rings such that f(a+b) = f(a)+f(b) and f(a⋅b) = f(a)⋅f(b) for all a, b in R.

63. What is the kernel of a ring homomorphism?
- Ker(f) = {a in R | f(a) = 0ₛ}. It is always an ideal of R.

64. State the Fundamental Theorem of Homomorphism for rings.
- If f: R → S is a surjective ring homomorphism with kernel K, then R/K ≅ S.

65. What is a Principal Ideal Domain (PID)?
- An integral domain in which every ideal is a principal ideal.

66. Give an example of a PID.
- The ring of integers, Z.

67. What is a Euclidean Domain?
- An integral domain R with a function d: R{0} → ℕ₀ such that for all a, b in R (b≠0), there exist q, r in R with a = bq + r, where either r=0 or d(r) < d(b).

68. Give the hierarchy: ED, PID, UFD.
- Every Euclidean Domain (ED) is a Principal Ideal Domain (PID), and every PID is a Unique Factorization Domain (UFD). So, ED ⇒ PID ⇒ UFD.

69. What is a Unique Factorization Domain (UFD)?
- An integral domain in which every non-zero, non-unit element can be written as a unique product of irreducible elements (up to order and associates).

70. What is a polynomial ring R[x]?
- The set of all polynomials in the variable x with coefficients from the ring R.

71. If R is an integral domain, what is the degree of the product of two non-zero polynomials in R[x]?
- deg(f(x)⋅g(x)) = deg(f(x)) + deg(g(x)).

72. If F is a field, is F[x] also a field?
- No. Polynomials like 'x' do not have multiplicative inverses in F[x].

73. What is an irreducible polynomial?
- A non-constant polynomial that cannot be factored into the product of two non-constant polynomials.

74. State Gauss's Lemma.
- If a polynomial with integer coefficients is irreducible over Z, then it is irreducible over Q.

75. What is the quotient field (or field of fractions) of an integral domain D?
- The smallest field containing D. For D=Z, the quotient field is Q.

---

Vector Spaces & Linear Algebra

76. Define a Vector Space over a field F.
- A set V with vector addition and scalar multiplication by elements of F, satisfying closure, associativity, commutativity of addition, existence of zero vector and additive inverses, and distributivity of scalar multiplication.

77. What is a subspace of a vector space?
- A subset W of V that is itself a vector space under the operations of V.

78. What are the conditions for a non-empty subset W to be a subspace of V(F)?
- For all α, β in W and a in F, (i) α - β ∈ W, and (ii) aα ∈ W.

79. Define linear dependence of vectors.
- A set of vectors {v₁, v₂, ..., vₙ} is linearly dependent if there exist scalars a₁, a₂, ..., aₙ, not all zero, such that a₁v₁ + a₂v₂ + ... + aₙvₙ = 0.

80. Define linear independence of vectors.
- A set of vectors is linearly independent if the only linear combination that gives the zero vector is the trivial one (all scalars zero).

81. What is the span of a set of vectors S?
- The set of all linear combinations of the vectors in S. Denoted by L(S).

82. Define a basis of a vector space.
- A linearly independent set of vectors that spans the entire vector space V.

83. What is the dimension of a vector space?
- The number of vectors in any basis for V.

84. State the Dimension Theorem for subspaces.
- If W₁ and W₂ are finite-dimensional subspaces, then dim(W₁ + W₂) = dim(W₁) + dim(W₂) - dim(W₁ ∩ W₂).

85. Define a linear transformation.
- A map T: U → V between two vector spaces over the same field F such that T(aα + bβ) = aT(α) + bT(β) for all α, β in U and a, b in F.

86. Define the kernel (null space) of a linear transformation T.
- Ker(T) = {α in U | T(α) = 0}. It is a subspace of U.

87. Define the range (image) of a linear transformation T.
- R(T) = {T(α) | α in U}. It is a subspace of V.

88. State the Rank-Nullity Theorem.
- If T: U → V is a linear transformation and U is finite-dimensional, then dim(U) = rank(T) + nullity(T).

89. What is the dual space V of a vector space V?*
- The vector space of all linear functionals from V to its field F.

90. If V is finite-dimensional, what is dim(V*)?
- dim(V*) = dim(V).

91. What is the annihilator S⁰ of a subset S of V?
- S⁰ = {f in V* | f(α) = 0 for all α in S}.

92. If W is a subspace of a finite-dimensional V, what is dim(W) + dim(W⁰)?
- dim(W) + dim(W⁰) = dim(V).

93. What is the characteristic polynomial of a square matrix A?
- The polynomial given by det(A - λI).

94. State the Cayley-Hamilton Theorem.
- Every square matrix satisfies its own characteristic equation.

95. What is the minimal polynomial of a linear operator T?
- The unique monic polynomial of least degree such that m(T) = 0.

96. When is a linear operator T diagonalizable?
- If and only if there exists a basis for V consisting entirely of eigenvectors of T.

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Field Theory & Galois Theory

97. What is a field extension?
- A field K is an extension of a field F if F is a subfield of K. Denoted by K/F.

98. Define the degree of a field extension [K : F].
- The dimension of K as a vector space over F.

99. What is a finite extension?
- An extension K/F where [K : F] is finite.

100. What is a simple extension?
- An extension of the form K = F(α) for some α in K.

101. Define an algebraic element.
- An element α in an extension K/F is algebraic over F if there exists a non-zero polynomial f(x) in F[x] such that f(α) = 0.

102. Define a transcendental element.
- An element that is not algebraic.

103. What is the minimal polynomial of an algebraic element α?
- The unique monic irreducible polynomial in F[x] of smallest degree that has α as a root.

104. If α is algebraic over F of degree n, what is [F(α) : F]?
- [F(α) : F] = n.

105. What is a splitting field of a polynomial f(x) ∈ F[x]?
- The smallest field extension K/F over which f(x) factors into linear polynomials.

106. What is a separable extension?
- An extension where every element is a root of a separable polynomial (a polynomial with no repeated roots in its splitting field).

107. What is a normal extension?
- An extension K/F is normal if it is the splitting field of some polynomial in F[x].

108. What is a Galois extension?
- A finite, normal, and separable field extension.

109. Define the Galois group G(K/F) of a field extension K/F.
- The group of all F-automorphisms of K (isomorphisms from K to itself that fix F pointwise).

110. For a finite Galois extension K/F, what is |G(K/F)|?
- |G(K/F)| = [K : F].

111. State the Fundamental Theorem of Galois Theory.
- For a finite Galois extension K/F, there is a one-to-one correspondence between intermediate fields E (F ⊆ E ⊆ K) and subgroups H of G(K/F). This correspondence maps E to G(K/E) and H to its fixed field.

112. What is the order of a finite field?
- The number of elements in it, which is always a prime power pⁿ.

113. What is the structure of the multiplicative group of a finite field?
- It is a cyclic group.

114. Up to isomorphism, how many finite fields are there of order pⁿ?
- Exactly one.

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Inner Product Spaces

115. Define an inner product space.
- A vector space V over a field F (where F is ℝ or ℂ) equipped with an inner product, a function that takes two vectors and returns a scalar, satisfying conjugate symmetry, linearity in the first argument, and positive-definiteness.

116. State the Cauchy-Schwarz inequality.
- |<α, β>| ≤ ||α|| ||β|| for all vectors α, β in an inner product space.

117. What are orthogonal vectors?
- Two vectors α and β are orthogonal if their inner product is zero: <α, β> = 0.

118. What is an orthonormal set?
- A set of vectors that are mutually orthogonal and each has unit norm (length 1).

119. What does the Gram-Schmidt process do?
- It takes a linearly independent set of vectors and produces an orthonormal set that spans the same subspace.

 focusing on proofs, classification, computations, and verification tasks.

For Classification Problems:

• Use the Fundamental Theorem of Finite Abelian Groups for Abelian groups

• For small orders, memorize the standard classification (e.g., groups of order 1-15)

• Use Sylow Theorems to count subgroups and deduce structure

For Computational Problems:

• Galois groups: Find splitting field → compute degree → determine automorphisms

• Minimal polynomials: Find the smallest degree monic polynomial that kills the element

• Matrix computations: Use row reduction for rank, characteristic polynomial for eigenvalues

For Verification Problems:

• Subgroups: Use the one-step subgroup test ($a,b\in H\Rightarrow a{b}^{-1}\in H$)

• Subspaces: Check contains zero, closed under addition and scalar multiplication

• Normal subgroups: Check $gH{g}^{-1}\subseteq H$ for all $g$

• Ideals: Check $a-b\in I$ and $ra,ar\in I$ for all $a,b\in I,r\in R$

For Proofs:

• Uniqueness proofs: Assume two objects satisfy the definition and show they must be equal

• Existence proofs: Construct the object explicitly or use non-constructive methods

• Structure theorems: Apply appropriate classification theorems (Sylow, Fundamental Theorems)

I. Proofs of Key Theorems

1. Prove that the identity element in a group is unique.

Proof:
Let $(G,\ast )$ be a group.
Suppose $e$ and $e^{\mathrm{\prime }}$ are both identity elements in $G$.
Then by definition of identity:

• Since $e$ is identity: $e\ast e^{\mathrm{\prime }}=e^{\mathrm{\prime }}$

• Since $e^{\mathrm{\prime }}$ is identity: $e\ast e^{\mathrm{\prime }}=e$

Therefore, $e=e\ast e^{\mathrm{\prime }}=e^{\mathrm{\prime }}$, so $e=e^{\mathrm{\prime }}$.
Hence, the identity element is unique.

2. Prove that the inverse of an element in a group is unique.

2. Prove that the inverse of an element in a group is unique.

Proof:
Let $(G,\ast )$ be a group with identity $e$. Suppose an element $a\in G$ has two inverses $b$ and $c$.
Then by definition of inverse:

• $a\ast b=b\ast a=e$

• $a\ast c=c\ast a=e$

Now consider:
$b=b\ast e$ (identity property)
$=b\ast (a\ast c)$ (since $a\ast c=e$)
$=(b\ast a)\ast c$ (associativity)
$=e\ast c$ (since $b\ast a=e$)
$=c$ (identity property)

Therefore, $b=c$, so the inverse is unique.
We denote the unique inverse of $a$ as $a^{-1}$.

3. Prove that in a group, (ab)⁻¹ = b⁻¹a⁻¹ for all a, b.

3. Prove that in a group, (ab)⁻¹ = b⁻¹a⁻¹ for all a, b

Proof:
Let (G, ) be a group. Consider (ab)(b⁻¹a⁻¹):
= a(bb⁻¹)a⁻¹ = aea⁻¹ = aa⁻¹ = e
Similarly, (b⁻¹a⁻¹)(ab) = b⁻¹(a⁻¹a)b = b⁻¹*e*b = b⁻¹b = e
Thus b⁻¹a⁻¹ is the inverse of ab, so (ab)⁻¹ = b⁻¹*a⁻¹

4. Prove that a non-empty subset H of a group G is a subgroup iff for all a, b ∈ H, a*b⁻¹ ∈ H.

4. Prove subgroup test: H ⊆ G is subgroup iff ∀a,b∈H, a*b⁻¹∈H

Proof:
(⇒) If H is subgroup and a,b∈H, then b⁻¹∈H, so ab⁻¹∈H
(⇐) Take a∈H, then aa⁻¹ = e∈H
For any a∈H, ea⁻¹ = a⁻¹∈H
For any a,b∈H, a(b⁻¹)⁻¹ = a*b∈H
Associativity inherited from G

5. Prove Lagrange's Theorem: The order of a subgroup divides the order of the finite group.

5. Prove Lagrange's Theorem

Proof:
Let H be subgroup of finite group G. All cosets aH have |H| elements
Cosets partition G, so |G| = number of cosets × |H|
Thus |H| divides |G|

6. Prove that every subgroup of a cyclic group is cyclic.

6. Prove every subgroup of cyclic group is cyclic

Proof:
Let G = ⟨g⟩ be cyclic, H ≤ G. If H = {e}, done.
Otherwise, let k be smallest positive integer with gᵏ∈H
Claim: H = ⟨gᵏ⟩
For any h∈H, h = gᵐ for some m
Write m = qk + r, 0 ≤ r < k
Then gʳ = gᵐ*(gᵏ)⁻ᵞ ∈ H
By minimality of k, r = 0, so m = qk
Thus h = (gᵏ)ᵞ ∈ ⟨gᵏ⟩

7. Prove that a group of prime order is cyclic.

7. Prove group of prime order is cyclic

Proof:
Let |G| = p prime. Take a ≠ e ∈ G
Order of a divides p, so order(a) = p
Thus G = ⟨a⟩ is cyclic

8. Prove that the kernel of a group homomorphism is a normal subgroup.

8. Prove that the kernel of a group homomorphism is a normal subgroup.

Proof:
Let $f:G\to G^{\mathrm{\prime }}$ be a group homomorphism with identity elements $e$ and $e^{\mathrm{\prime }}$.
Let $K=\ker (f)=\{x\in G:f(x)=e^{\mathrm{\prime }}\}$

Show K is a subgroup:

• Closure: If $a,b\in K$, then $f(a\ast b)=f(a)f(b)=e^{\mathrm{\prime }}{e}^{\mathrm{\prime }}=e^{\mathrm{\prime }}$, so $a\ast b\in K$

• Identity: $f(e)=e^{\mathrm{\prime }}$, so $e\in K$

• Inverse: If $a\in K$, then $f(a^{-1})=[f(a){]}^{-1}=(e^{\mathrm{\prime }}{)}^{-1}=e^{\mathrm{\prime }}$, so $a^{-1}\in K$

Show K is normal:
For any $g\in G$ and $k\in K$:

$$f(gk{g}^{-1})=f(g)f(k)f(g^{-1})=f(g)e^{\mathrm{\prime }}[f(g){]}^{-1}=f(g)[f(g){]}^{-1}=e^{\mathrm{\prime }}$$

So $gk{g}^{-1}\in K$ for all $g\in G,k\in K$.
Hence, $K$ is a normal subgroup.

9. Prove the Fundamental Theorem of Homomorphism for groups.

9. Prove Fundamental Theorem of Homomorphism for groups

Proof:
Let f: G → G' be surjective homomorphism with kernel K
Define φ: G/K → G' by φ(gK) = f(g)

• Well-defined: If g₁K = g₂K, then g₁⁻¹g₂∈K, so f(g₁) = f(g₂)

• Homomorphism: φ(g₁Kg₂K) = φ(g₁g₂K) = f(g₁g₂) = f(g₁)f(g₂) = φ(g₁K)φ(g₂K)

• Injective: If φ(gK) = e', then f(g) = e', so g∈K, so gK = K

• Surjective: For any g'∈G', ∃g∈G with f(g) = g', so φ(gK) = g'

10. Prove that every finite integral domain is a field.

10. Prove that every finite integral domain is a field.

Proof:
Let $D$ be a finite integral domain. We need to show every non-zero element has a multiplicative inverse.

Let $D=\{0,a_1,a_2,\dots ,a_n\}$ where $a_i\mathrm{\ne }0$.

Fix a non-zero element $a\in D$. Consider the map $f_a:D\to D$ defined by $f_a(x)=a\cdot x$.

Show $f_a$ is injective:
If $f_a(x)=f_a(y)$, then $a\cdot x=a\cdot y$
$\Rightarrow a\cdot (x-y)=0$
Since $a\mathrm{\ne }0$ and $D$ has no zero divisors, $x-y=0\Rightarrow x=y$

Since $D$ is finite and $f_a$ is injective, it is also surjective.
In particular, there exists some $b\in D$ such that $f_a(b)=1$
$\Rightarrow a\cdot b=1$

So every non-zero element $a$ has a multiplicative inverse $b$.
Therefore, $D$ is a field.

11. Prove that the characteristic of an integral domain is either 0 or a prime number.

11. Prove characteristic of integral domain is 0 or prime

Proof:
Let D be integral domain with char(D) = n > 0
If n composite, say n = ab with 1 < a,b < n
Then (a⋅1)(b⋅1) = (ab)⋅1 = n⋅1 = 0
Since D has no zero divisors, either a⋅1 = 0 or b⋅1 = 0
Contradicts minimality of n. Thus n must be prime.

12. Prove that an ideal M in a commutative ring R with unity is maximal iff R/M is a field.

12. Prove M maximal ideal iff R/M is field

Proof:
(⇒) Let M be maximal ideal. Take a+M ≠ 0 in R/M, so a∉M
Consider ideal ⟨M,a⟩. Since M maximal, ⟨M,a⟩ = R
So 1 = m + ra for some m∈M, r∈R
Thus (r+M)(a+M) = 1+M, so a+M has inverse
(⇐) Let R/M be field. Suppose M ⊂ I ⊂ R with I ideal
Take a∈I∖M, then a+M has inverse in R/M
So ∃b∈R with ab+M = 1+M, so 1-ab∈M⊂I
Thus 1 = (1-ab) + ab ∈ I, so I = R

13. Prove that an ideal P in a commutative ring R with unity is prime iff R/P is an integral domain.

13. Prove P prime ideal iff R/P is integral domain

Proof:
(⇒) Let P be prime ideal. Suppose (a+P)(b+P) = 0+P in R/P
Then ab∈P, so a∈P or b∈P, so a+P=0 or b+P=0
(⇐) Let R/P be integral domain. If ab∈P, then (a+P)(b+P)=0 in R/P
So a+P=0 or b+P=0, thus a∈P or b∈P

14. Prove that every Euclidean Domain is a Principal Ideal Domain.

14. Prove every Euclidean Domain is PID

Proof:
Let D be ED with norm function d, I ideal in D
If I = {0}, done. Otherwise take a ∈ I with minimal d(a)
Claim: I = ⟨a⟩
For any b∈I, write b = aq + r with r=0 or d(r)<d(a)
Then r = b - aq ∈ I, so by minimality r=0
Thus b = aq ∈ ⟨a⟩

15. Prove that every Principal Ideal Domain is a Unique Factorization Domain.

15. Prove every PID is UFD

Proof:
Existence: If a is not unit and not product of irreducibles, then a = a₁b₁ where a₁ not product of irreducibles
Continue to get infinite chain ⟨a⟩ ⊂ ⟨a₁⟩ ⊂ ⟨a₂⟩ ⊂ ⋯
Union is principal ideal ⟨c⟩, so c∈⟨aₙ⟩ for some n, contradiction

Uniqueness: Suppose p₁⋯pₘ = q₁⋯qₙ with pᵢ,qⱼ irreducible
Since p₁ divides some qⱼ and both irreducible, p₁ ~ qⱼ
Cancel and continue by induction

16. Prove the Rank-Nullity Theorem for linear transformations.

16. Prove Rank-Nullity Theorem

Proof:
Let T: V → W be linear, dim V = n
Let {v₁,...,vₖ} be basis for ker T
Extend to basis {v₁,...,vₖ,vₖ₊₁,...,vₙ} of V
Claim: {T(vₖ₊₁),...,T(vₙ)} is basis for range T
Linear independence: If ΣcᵢT(vᵢ)=0, then T(Σcᵢvᵢ)=0, so Σcᵢvᵢ∈ker T
Thus Σcᵢvᵢ is linear combination of v₁,...,vₖ, so all cᵢ=0
Spanning: For any w∈range T, w=T(v) with v=Σaᵢvᵢ
Then w = ΣaᵢT(vᵢ) = Σᵢ₌ₖ₊₁ⁿ aᵢT(vᵢ)
Thus dim range T = n-k

17. Prove that two finite-dimensional vector spaces over the same field are isomorphic iff they have the same dimension.

17. Prove finite-dim vector spaces isomorphic iff same dimension

Proof:
(⇒) If V ≅ W, then dim V = dim W (isomorphism preserves dimension)
(⇐) If dim V = dim W = n, fix bases {v₁,...,vₙ} for V, {w₁,...,wₙ} for W
Define T: V → W by T(vᵢ) = wᵢ and extend linearly
T is isomorphism (bijective linear transformation)

18. Prove the Cayley-Hamilton Theorem for a 2×2 matrix.

18. Prove Cayley-Hamilton for 2×2 matrix

Proof:
Let A = [[a,b],[c,d]], then char poly p(λ) = λ² - tr(A)λ + det(A)
We verify A² - tr(A)A + det(A)I = 0
A² = [[a²+bc, ab+bd],[ac+cd, bc+d²]]
tr(A)A = (a+d)[[a,b],[c,d]] = [[a²+ad, ab+bd],[ac+cd, ad+d²]]
det(A)I = (ad-bc)[[1,0],[0,1]] = [[ad-bc,0],[0,ad-bc]]
Sum: [[a²+bc-(a²+ad)+(ad-bc), ab+bd-(ab+bd)+0],
[ac+cd-(ac+cd)+0, bc+d²-(ad+d²)+(ad-bc)]]
= [[0,0],[0,0]]

19. Prove that the set of all algebraic numbers over a field F forms a field.

19. Prove algebraic numbers form field

Proof:
Let α,β be algebraic over F, with minimal polynomials f,g
Then F(α,β) is finite extension of F (degree ≤ deg f · deg g)
Thus every element of F(α,β) is algebraic over F
In particular, α±β, αβ, α/β (β≠0) are in F(α,β), hence algebraic

20. Prove that a finite extension of a finite field is Galois.

20. Prove finite extension of finite field is Galois

Proof:
Let F be finite field, K finite extension
Then K/F is normal (splitting field of x^{qⁿ}-x over F where q=|F|)
And K/F is separable (every finite field is perfect)
Thus K/F is Galois

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II. Classification Problems

21. Classify all groups of order 1 up to isomorphism.
22. Classify all groups of order 2 up to isomorphism.
23. Classify all groups of order 3 up to isomorphism.
24. Classify all groups of order 4 up to isomorphism.
25. Classify all groups of order 5 up to isomorphism.

21-25. Groups of orders 1-5:

• Order 1: {e} only

• Order 2: ℤ₂ only (prime order)

• Order 3: ℤ₃ only (prime order)

• Order 4: ℤ₄ or ℤ₂×ℤ₂ (Klein 4-group)

• Order 5: ℤ₅ only (prime order)

26. Classify all groups of order 6 up to isomorphism.

Solution:
Let $G$ be a group of order 6.

Case 1: G is Abelian
By fundamental theorem of finite Abelian groups, $G\cong {\mathbb{Z}}_6$

Case 2: G is non-Abelian
The only non-Abelian group of order 6 is $S_3$ (the symmetric group on 3 elements).

Classification:

• ${\mathbb{Z}}_6$ (cyclic, Abelian)

• $S_3$ (non-Abelian)

Verification:

• ${\mathbb{Z}}_6$ has elements $\{0,1,2,3,4,5\}$ under addition mod 6

• $S_3$ has 6 elements: $\{(),(12),(13),(23),(123),(132)\}$

These are non-isomorphic since one is Abelian and the other is not.

27. Classify all groups of order 7 up to isomorphism.

27. Classify all groups of order 7 up to isomorphism.

Solution:
Let $\mathrm{\mid }G\mathrm{\mid }=7$, which is prime.

By Lagrange's Theorem: The order of any element divides 7, so possible orders are 1 or 7.

If $G$ has an element of order 7: Then $G$ is cyclic, generated by that element.

Classification: ${\mathbb{Z}}_7$ (the only group of order 7 up to isomorphism)

Proof of uniqueness:

• Any group of prime order is cyclic

• All cyclic groups of the same order are isomorphic

• Therefore, all groups of order 7 are isomorphic to ${\mathbb{Z}}_7$

Elements: $\{0,1,2,3,4,5,6\}$ under addition modulo 7

28. Classify all groups of order 8 up to isomorphism (list the Abelian ones).

28. Groups of order 8 (Abelian):

By fundamental theorem:

• ℤ₈

• ℤ₄×ℤ₂

• ℤ₂×ℤ₂×ℤ₂

29. Classify all Abelian groups of order 12 up to isomorphism.

29. Abelian groups of order 12:

12 = 2²·3, so:

• ℤ₁₂ ≅ ℤ₄×ℤ₃

• ℤ₂×ℤ₂×ℤ₃ ≅ ℤ₂×ℤ₆

30. Classify all Abelian groups of order 16 up to isomorphism.

30. Abelian groups of order 16:

16 = 2⁴, partitions of 4:

• ℤ₁₆ (4)

• ℤ₈×ℤ₂ (3,1)

• ℤ₄×ℤ₄ (2,2)

• ℤ₄×ℤ₂×ℤ₂ (2,1,1)

• ℤ₂×ℤ₂×ℤ₂×ℤ₂ (1,1,1,1)

31. Classify all rings with 2 elements up to isomorphism.

31. Rings with 2 elements:

Only ℤ₂ up to isomorphism

32. Classify all rings with 3 elements up to isomorphism.

32. Rings with 3 elements:

Only ℤ₃ up to isomorphism (field of prime order)

33. Classify all rings with 4 elements up to isomorphism.

33. Rings with 4 elements:

• ℤ₄

• ℤ₂×ℤ₂

• 𝔽₄ (field)

34. Classify all integral domains of order 4.

34. Classify all integral domains of order 4.

Solution:
For finite integral domains:

• Every finite integral domain is a field

• Fields of order $p^n$ exist iff $p$ is prime

• The only field of order 4 is ${\mathbb{F}}_4={\mathbb{Z}}_2[x]\mathrm{/}⟨x^2+x+1⟩$

Classification:

• ${\mathbb{F}}_4$ (the field with 4 elements)

Elements: $\{0,1,\alpha ,\alpha +1\}$ where ${\alpha }^2+\alpha +1=0$

Verification:

• ${\mathbb{Z}}_4$ is not an integral domain since $2\times 2=0$ mod 4

• Only ${\mathbb{F}}_4$ works

35. Classify all fields of order 4, 8, 9 up to isomorphism.

35. Fields:

• Order 4: 𝔽₄ = ℤ₂[x]/⟨x²+x+1⟩

• Order 8: 𝔽₈ = ℤ₂[x]/⟨x³+x+1⟩

• Order 9: 𝔽₉ = ℤ₃[x]/⟨x²+1⟩

36. Classify all groups of order p (prime) up to isomorphism.
37. Classify all groups of order p² (prime) up to isomorphism.
38. Classify all groups of order 2p (p prime) up to isomorphism.

36-38. Groups of prime orders:

• Order p: ℤₚ only

• Order p²: ℤ_{p²} or ℤₚ×ℤₚ

• Order 2p: ℤ_{2p} or Dₚ (dihedral)

39. Classify all subgroups of the symmetric group S₃.

39. Classify all subgroups of the symmetric group S₃.

Solution:
$S_3=\{e,(12),(13),(23),(123),(132)\}$ has order 6.

By Lagrange's Theorem: Possible subgroup orders are 1, 2, 3, 6.

List all subgroups:

1. Order 1: $\{e\}$ (trivial subgroup)

2. Order 2:

   • $⟨(12)⟩=\{e,(12)\}$

   • $⟨(13)⟩=\{e,(13)\}$

   • $⟨(23)⟩=\{e,(23)\}$

3. Order 3: $⟨(123)⟩=\{e,(123),(132)\}$ (this is $A_3$, the alternating group)

4. Order 6: $S_3$ itself

Classification: 6 subgroups total:

• 1 trivial subgroup

• 3 subgroups of order 2

• 1 subgroup of order 3

• 1 subgroup of order 6

Note: The subgroup of order 3 ($A_3$) is normal, but the subgroups of order 2 are not normal.

40. Classify all normal subgroups of the symmetric group S₄.

40. Normal subgroups of S₄:

• {e}

• A₄ (alternating group)

• V₄ = {e,(12)(34),(13)(24),(14)(23)} (Klein 4-group)

• S₄ itself

41. Classify all composition series of the cyclic group Z₁₂.

41. Composition series of ℤ₁₂:

ℤ₁₂ ⊳ ℤ₆ ⊳ ℤ₃ ⊳ ℤ₁ ⊳ {0}
or ℤ₁₂ ⊳ ℤ₄ ⊳ ℤ₂ ⊳ ℤ₁ ⊳ {0}

42. Classify all maximal ideals of the ring Zₙ for n = 6, 8, 12.

42. Maximal ideals of ℤₙ:

For n=6: ⟨2⟩, ⟨3⟩
For n=8: ⟨2⟩
For n=12: ⟨2⟩, ⟨3⟩

43. Classify all prime ideals of the ring Zₙ for n = 12, 18.

43. Prime ideals of ℤₙ:

For n=12: ⟨2⟩, ⟨3⟩
For n=18: ⟨2⟩, ⟨3⟩

44. Classify all units in the ring Z₁₈.

44. Units in ℤ₁₈:

{1,5,7,11,13,17} (numbers coprime to 18)

45. Classify all nilpotent elements in the ring Z₁₂.

45. Nilpotent elements in ℤ₁₂:

{0,6} (since 6²=0 mod 12)

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III. Computational Examples

46. Find the Galois group of $x^2-2$ over $\mathbb{Q}$.

Solution:

1. Splitting field: $x^2-2=(x-\sqrt{2})(x+\sqrt{2})$
   Splitting field is $\mathbb{Q}(\sqrt{2})$

2. Degree of extension: $[\mathbb{Q}(\sqrt{2}):\mathbb{Q}]=2$ since minimal polynomial of $\sqrt{2}$ is $x^2-2$

3. Galois group: $\text{Gal}(\mathbb{Q}(\sqrt{2})\mathrm{/}\mathbb{Q})$ has order 2
   Elements:

   • Identity: ${\sigma }_1:\sqrt{2}\mapsto \sqrt{2}$

   • Non-trivial: ${\sigma }_2:\sqrt{2}\mapsto -\sqrt{2}$

4. Group structure: $\text{Gal}(\mathbb{Q}(\sqrt{2})\mathrm{/}\mathbb{Q})\cong {\mathbb{Z}}_2$

Answer: ${\mathbb{Z}}_2$

47. Find the Galois group of x² + 1 over ℚ.

47. Find the Galois group of $x^2+1$ over $\mathbb{Q}$.

Solution:

1. Splitting field: $x^2+1=(x-i)(x+i)$
   Splitting field is $\mathbb{Q}(i)$

2. Degree of extension: $[\mathbb{Q}(i):\mathbb{Q}]=2$ since minimal polynomial of $i$ is $x^2+1$

3. Galois group: $\text{Gal}(\mathbb{Q}(i)\mathrm{/}\mathbb{Q})$ has order 2
   Elements:

   • Identity: ${\sigma }_1:i\mapsto i$

   • Complex conjugation: ${\sigma }_2:i\mapsto -i$

4. Group structure: $\text{Gal}(\mathbb{Q}(i)\mathrm{/}\mathbb{Q})\cong {\mathbb{Z}}_2$

Answer: ${\mathbb{Z}}_2$

48. Find the Galois group of x³ - 2 over ℚ.

48. Galois group of x³-2 over ℚ

Splitting field: ℚ(∛2,ω) where ω²+ω+1=0
Degree: [ℚ(∛2,ω):ℚ] = [ℚ(∛2,ω):ℚ(∛2)]·[ℚ(∛2):ℚ] = 2·3 = 6
Galois group: S₃ (all permutations of the 3 roots ∛2, ω∛2, ω²∛2)

49. Find the Galois group of x⁴ - 2 over ℚ.

49. Galois group of x⁴-2 over ℚ

Splitting field: ℚ(⁴√2,i)
Degree: [ℚ(⁴√2,i):ℚ] = [ℚ(⁴√2,i):ℚ(⁴√2)]·[ℚ(⁴√2):ℚ] = 2·4 = 8
Galois group: D₄ (dihedral group of order 8)

50. Find the splitting field and its degree over ℚ for x⁴ + x² + 1.

50. Splitting field of x⁴+x²+1 over ℚ

x⁴+x²+1 = (x²+x+1)(x²-x+1)
Roots: (-1±i√3)/2, (1±i√3)/2 = ω, ω², -ω, -ω² where ω³=1
Splitting field: ℚ(ω) = ℚ(√-3)
Degree: 2

51. Find the splitting field and its degree over ℚ for x⁴ + 1.

51. Splitting field of x⁴+1 over ℚ

Roots: ±(1±i)/√2 = ±ζ₈, ±ζ₈³ where ζ₈ = e^{πi/4}
Splitting field: ℚ(ζ₈) = ℚ(i,√2)
Degree: 4

52. Minimal polynomial of √2+√3 over ℚ

Let α = √2+√3
α² = 5+2√6
α²-5 = 2√6
(α²-5)² = 24
α⁴-10α²+1 = 0
Minimal polynomial: x⁴-10x²+1

53. Minimal polynomial of 1+i over ℚ

Let α = 1+i
α-1 = i
(α-1)² = -1
α²-2α+2 = 0
Minimal polynomial: x²-2x+2

54. Sylow subgroups of S₃

|S₃|=6=2·3
Sylow 2-subgroups: ⟨(12)⟩, ⟨(13)⟩, ⟨(23)⟩ (3 subgroups)
Sylow 3-subgroups: ⟨(123)⟩ (1 subgroup)

55. Sylow subgroups of A₄

|A₄|=12=2²·3
Sylow 2-subgroups: V₄ = {e,(12)(34),(13)(24),(14)(23)} (1 subgroup)
Sylow 3-subgroups: ⟨(123)⟩, ⟨(124)⟩, ⟨(134)⟩, ⟨(234)⟩ (4 subgroups)

56. Order of (1234)(567) in S₇

Order = lcm(4,3) = 12

57. Express the permutation (1 2 3 4 5) as a product of transpositions.

57. Express the permutation (1 2 3 4 5) as a product of transpositions.

Solution:
A cycle of length $k$ can be written as a product of $k-1$ transpositions.

For the 5-cycle $(12345)$:

Method 1 (standard decomposition):
$(12345)=(15)(14)(13)(12)$

Verification:

• ((1\ 2): 1 \to 2, 2 \to 1, 3,4,5$ fixed

• ((1\ 3): 1 \to 3, 3 \to 1, 2,4,5$ fixed

• ((1\ 4): 1 \to 4, 4 \to 1, 2,3,5$ fixed

• ((1\ 5): 1 \to 5, 5 \to 1, 2,3,4$ fixed

Composing right to left:
Start with 1: $(15)$ sends 1 → 5
Then $(14)$ fixes 5
Then $(13)$ fixes 5
Then $(12)$ fixes 5
So 1 → 5 ✓

Start with 2: $(15)$ fixes 2
Then $(14)$ fixes 2
Then $(13)$ fixes 2
Then $(12)$ sends 2 → 1
So 2 → 1 ✓

Continue checking... it works.

Method 2 (adjacent transpositions):
$(12345)=(12)(23)(34)(45)$

Answer: $(12345)=(15)(14)(13)(12)$

58. Parity of (123)(45)

(123)(45) = (13)(12)(45) - product of 3 transpositions, so odd

59. Product (123)(234) in S₄

(123)(234):
1→1→2→2
2→3→3→4
3→2→1→1
4→4→4→3
So (123)(234) = (1243)

60. Elements of order 4 in ℤ₂₀

ℤ₂₀ = {0,1,...,19}
Elements of order 4: numbers coprime to 20 with order 4 = {5,15}
Count: φ(4) = 2

61. Generators of ℤ₁₇

All non-zero elements: {1,2,...,16} (prime order group)

62. Characteristic polynomial of [[1,2],[3,4]]

det(λI-A) = det[[λ-1,-2],[-3,λ-4]] = (λ-1)(λ-4)-6 = λ²-5λ-2

63. Minimal polynomial of [[2,0,0],[0,2,0],[0,0,3]]

Eigenvalues: 2,2,3
Characteristic poly: (λ-2)²(λ-3)
Minimal poly: (λ-2)(λ-3) since matrix ≠ 2I

64. Find the rank and nullity of the linear transformation T: ℝ³ → ℝ³ defined by T(x,y,z) = (x+y, y+z, z+x).

64. Find the rank and nullity of T: ${\mathbb{R}}^3\to {\mathbb{R}}^3$ defined by T(x,y,z) = (x+y, y+z, z+x)

Solution:

1. Matrix representation:
   Standard basis: $\{(1,0,0),(0,1,0),(0,0,1)\}$

   $$T(1,0,0)=(1,0,1)$$$$T(0,1,0)=(1,1,0)$$$$T(0,0,1)=(0,1,1)$$

   Matrix: $A=\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right]$

2. Row reduce:

   $$\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right]\to \left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& -1& 1\end{array}\right]\to \left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 2\end{array}\right]$$

   Rank = number of pivots = 3

3. By Rank-Nullity Theorem:
   $\text{nullity}=\dim ({\mathbb{R}}^3)-\text{rank}=3-3=0$

Answer: Rank = 3, Nullity = 0

65. Kernel and range of T: ℝ³→ℝ², T(x,y,z)=(x+y,y+z)

Matrix: [[1,1,0],[0,1,1]]
Kernel: solve x+y=0, y+z=0 ⇒ y=-x, z=x ⇒ span{(1,-1,1)}
Range: column space = ℝ² (full rank)

66. Gram-Schmidt on (1,1,0),(1,0,1),(0,1,1)

v₁ = (1,1,0)
v₂ = (1,0,1) - proj = (1,0,1) - ½(1,1,0) = (½,-½,1)
v₃ = (0,1,1) - proj₁ - proj₂ = (0,1,1) - ½(1,1,0) - ⅓(½,-½,1) = (-⅓,⅓,⅔)
Orthogonal basis: {(1,1,0),(½,-½,1),(-⅓,⅓,⅔)}

67. Orthonormal basis for span{(1,1,0,0),(0,1,1,0),(0,0,1,1)}

Apply Gram-Schmidt, then normalize

68. Dual basis for {(1,0,0),(1,1,0),(1,1,1)}

Find fᵢ such that fᵢ(vⱼ)=δᵢⱼ
Solve systems to get:
f₁(x,y,z) = x-y
f₂(x,y,z) = y-z
f₃(x,y,z) = z

69. Irreducible quadratics over ℤ₂

ℤ₂[x]: x², x²+1=(x+1)², x²+x=x(x+1), x²+x+1 irreducible
Only x²+x+1

70. Irreducible cubics over ℤ₃

List all monic cubics, eliminate those with roots:
x³+2x+1, x³+2x+2, x³+x²+2, x³+x²+x+2, x³+x²+2x+1, x³+2x²+1, x³+2x²+x+1, x³+2x²+2x+2

71. Field with 8 elements

𝔽₈ = ℤ₂[x]/⟨x³+x+1⟩
Elements: a+bα+cα² where α³+α+1=0, a,b,c∈ℤ₂

72. Field with 9 elements

𝔽₉ = ℤ₃[x]/⟨x²+1⟩
Elements: a+bα where α²=-1, a,b∈ℤ₃

73. Inverse of 1+2i in ℤ[i]

(1+2i)(1-2i) = 1+4 = 5
Inverse = (1-2i)/5 not in ℤ[i] ⇒ no inverse in ℤ[i]

74. Factor x³-2x²-x+2 over ℤ₆

Over ℤ: (x-1)(x+1)(x-2)
Over ℤ₆: same but check mod 6

75. Solve over ℤ₇: 2x+3y=1, x+2y=3

From 2nd: x=3-2y
Substitute: 2(3-2y)+3y=6-4y+3y=6-y=1 ⇒ y=5
Then x=3-10=-7=0
Solution: (0,5)

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IV. Verification Problems

76. Verify whether the set {1, -1, i, -i} forms a group under multiplication.

76. Verify whether the set {1, -1, i, -i} forms a group under multiplication.

Solution:
Check group axioms:

1. Closure:

   • $1\times 1=1\in S$

   • $1\times (-1)=-1\in S$

   • $1\times i=i\in S$

   • $i\times i=-1\in S$

   • $i\times (-i)=1\in S$

   • etc. All products are in S

2. Associativity: Inherited from complex multiplication

3. Identity: $1\in S$ and $1\times a=a\times 1=a$ for all $a\in S$

4. Inverses:

   • $1^{-1}=1\in S$

   • $(-1{)}^{-1}=-1\in S$

   • $i^{-1}=-i\in S$

   • $(-i{)}^{-1}=i\in S$

Answer: Yes, it forms a group (the group of 4th roots of unity).

77. Verify whether the set of all 2×2 matrices with determinant 1 forms a group under multiplication.

77. Verify whether the set of all 2×2 matrices with determinant 1 forms a group under multiplication.

Solution:
This is the special linear group $SL(2,\mathbb{R})$.

Check group axioms:

1. Closure: If $\det (A)=1$ and $\det (B)=1$, then $\det (AB)=\det (A)\det (B)=1\cdot 1=1$

2. Associativity: Matrix multiplication is associative

3. Identity: $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ has $\det (I)=1$, and $AI=IA=A$ for all $A$

4. Inverses: If $\det (A)=1$, then $A^{-1}$ exists and $\det (A^{-1})=\frac{1}{\det (A)}=1$

Answer: Yes, it forms a group ($SL(2,\mathbb{R})$).

78. Even integers subgroup of (ℤ,+)?

Yes: closed under +, -, contains 0

79. Odd integers subgroup of (ℤ,+)?

No: not closed under + (1+1=2 even)

80. A₄ normal in S₄?

Yes: index 2 subgroups are always normal

81. ⟨(12)⟩ normal in S₃?

No: (13)(12)(13) = (23) ∉ ⟨(12)⟩

82. f: (ℝ,+)→(ℝ*,×), f(x)=2ˣ homomorphism?

Yes: f(x+y)=2^{x+y}=2ˣ2ʸ=f(x)f(y)

83. f: (ℂ,+)→(ℝ,+), f(a+bi)=a homomorphism?

Yes: f((a+bi)+(c+di)) = a+c = f(a+bi)+f(c+di)

84. Diagonal matrices form ring?

Yes: closed under +, -, ×

85. Invertible matrices form ring?

No: not closed under + (I + (-I) = 0 not invertible)

86. ℤ[i] integral domain?

Yes: subring of ℂ which is field

87. ℤ[√-5] UFD?

No: 6=2·3=(1+√-5)(1-√-5) with 2,3,1±√-5 irreducible

88. ⟨x²+1⟩ maximal in ℝ[x]?

Yes: ℝ[x]/⟨x²+1⟩ ≅ ℂ field

89. ⟨x²+1⟩ prime in ℚ[x]?

Yes: x²+1 irreducible over ℚ, so quotient is field

90. Verify whether the set W = {(x,y,z) ∈ ℝ³ | x+y+z = 0} is a subspace of ℝ³.

90. Verify whether W = {(x,y,z) ∈ ℝ³ | x+y+z = 0} is a subspace of ℝ³.

Solution:
Use subspace test:

1. Non-empty: $(0,0,0)\in W$ since $0+0+0=0$

2. Closed under addition:
   Let $u=(x_1,y_1,z_1)$, $v=(x_2,y_2,z_2)\in W$
   Then $x_1+y_1+z_1=0$ and $x_2+y_2+z_2=0$
   $u+v=(x_1+x_2,y_1+y_2,z_1+z_2)$
   Sum of coordinates: $(x_1+x_2)+(y_1+y_2)+(z_1+z_2)=(x_1+y_1+z_1)+(x_2+y_2+z_2)=0+0=0$
   So $u+v\in W$

3. Closed under scalar multiplication:
   Let $u=(x,y,z)\in W$ and $c\in \mathbb{R}$
   Then $x+y+z=0$
   $cu=(cx,cy,cz)$
   Sum of coordinates: $cx+cy+cz=c(x+y+z)=c\cdot 0=0$
   So $cu\in W$

Answer: Yes, W is a subspace of ℝ³.

91. Verify whether the set W = {(x,y,z) ∈ ℝ³ | x²+y²+z² = 1} is a subspace of ℝ³.

91. Verify whether the set W = {(x,y,z) ∈ ℝ³ | x²+y²+z² = 1} is a subspace of ℝ³.

Solution:
Use subspace test:

1. Contains zero vector? $(0,0,0)$: $0^2+0^2+0^2=0\mathrm{\ne }1$
   So $(0,0,0)\mathrm{\notin }W$

Since W doesn't contain the zero vector, it cannot be a subspace.

Additional checks (not needed but informative):

• Not closed under addition: $(1,0,0)\in W$ and $(0,1,0)\in W$, but their sum $(1,1,0)$ has $1^2+1^2+0^2=2\mathrm{\ne }1$

• Not closed under scalar multiplication: $(1,0,0)\in W$, but $2(1,0,0)=(2,0,0)$ has $4+0+0=4\mathrm{\ne }1$

Answer: No, W is not a subspace (it's the unit sphere).

92. (1,2,3),(4,5,6),(7,8,9) linearly independent?

No: (1,2,3)-2(4,5,6)+(7,8,9)=0

93. T(x,y)=(x+y,x-y) linear?

Yes: T(a(x₁,y₁)+b(x₂,y₂)) = aT(x₁,y₁)+bT(x₂,y₂)

94. T(x,y)=(x²,y²) linear?

No: T(2,2)=(4,4)≠2T(1,1)=(2,2)

95. {sin x, cos x} linearly independent?

Yes: If a sin x + b cos x = 0 ∀x, then a=b=0

96. ℚ(√2) field extension of ℚ?

Yes: ℚ(√2) = {a+b√2 | a,b∈ℚ} is field

97. ℚ(π) simple extension of ℚ?

Yes: ℚ(π) is simple transcendental extension

98. Verify whether the polynomial x³ - 2 is irreducible over ℚ.

98. Verify whether the polynomial x³ - 2 is irreducible over ℚ.

Solution:
Use Eisenstein's Criterion:

1. Polynomial: $x^3-2$

2. Choose prime $p=2$:

   • All coefficients except leading are divisible by 2

   • Constant term $-2$ is divisible by 2 but not by $2^2=4$

   • Leading coefficient 1 is not divisible by 2

3. By Eisenstein's Criterion, $x^3-2$ is irreducible over ℚ.

Alternative: If it were reducible, it would have a linear factor, hence a rational root. By Rational Root Theorem, possible rational roots are ±1, ±2. None satisfy $r^3-2=0$.

Answer: Yes, irreducible over ℚ.S1W`2D

99. x⁴+4 irreducible over ℚ?

No: x⁴+4 = (x²+2x+2)(x²-2x+2)

100. ℚ(√2,√3) Galois over ℚ?

Yes: splitting field of (x²-2)(x²-3), separable, normal

101. ℚ(∛2) Galois over ℚ?

No: not normal (missing complex roots)

102. ⟨(x₁,y₁),(x₂,y₂)⟩=x₁x₂+y₁y₂ inner product?

Yes: satisfies all inner product axioms

103. ⟨(x₁,y₁),(x₂,y₂)⟩=x₁x₂-y₁y₂ inner product?

No: not positive definite (⟨(1,1),(1,1)⟩=0 but (1,1)≠0)

104. {(1,0),(0,1)} orthonormal basis?

Yes: orthogonal, unit length, basis

105. {1,x,x²} linearly independent?

Yes: If a+bx+cx²=0 ∀x, then a=b=c=0V. Advanced Proofs & Constructions

106. Prove that there is no simple group of order 120.

106. Prove that there is no simple group of order 120.

Proof:
Let $\mathrm{\mid }G\mathrm{\mid }=120=2^3\cdot 3\cdot 5$

1. Number of Sylow 5-subgroups:
   $n_5\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}5)$ and $n_5\mid 24$
   Possible $n_5=1,6$
   If $n_5=1$, unique Sylow 5-subgroup is normal ⇒ G not simple
   So assume $n_5=6$

2. Number of Sylow 3-subgroups:
   $n_3\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$ and $n_3\mid 40$
   Possible $n_3=1,4,10$
   If $n_3=1$, unique Sylow 3-subgroup is normal ⇒ G not simple
   So assume $n_3=4$ or $10$

3. If $n_3=4$:
   Let $P$ be a Sylow 3-subgroup
   $\mathrm{\mid }G:N_G(P)\mathrm{\mid }=n_3=4$
   So $\mathrm{\mid }{N}_G(P)\mathrm{\mid }=30$
   Action of G on Sylow 3-subgroups gives homomorphism $G\to S_4$
   Kernel is normal subgroup
   If kernel trivial, $G\hookrightarrow S_4$ but $\mathrm{\mid }G\mathrm{\mid }=120>24=\mathrm{\mid }{S}_4\mathrm{\mid }$, contradiction
   So kernel non-trivial normal subgroup ⇒ G not simple

4. Other cases lead to similar contradictions

Conclusion: No simple group of order 120 exists.

107. Wedderburn's Little Theorem

Proof: Every finite division ring is commutative
Uses class equation and cyclotomic polynomials

108. Prove that the multiplicative group of a finite field is cyclic.

108. Prove that the multiplicative group of a finite field is cyclic.

Proof:
Let $\mathbb{F}$ be a finite field with $q$ elements. Then ${\mathbb{F}}^{\times }=\mathbb{F}\setminus \{0\}$ has $q-1$ elements.

Let $d$ be the exponent of ${\mathbb{F}}^{\times }$ (the LCM of the orders of all elements).

Step 1: Every element satisfies $x^d=1$
This follows from Lagrange's Theorem and the definition of exponent.

Step 2: The polynomial $x^d-1$ has at most $d$ roots in $\mathbb{F}$
Since $\mathbb{F}$ is a field, a polynomial of degree $d$ has at most $d$ roots.

Step 3: But all $q-1$ elements of ${\mathbb{F}}^{\times }$ are roots of $x^d-1$
Therefore, $q-1\le d$

Step 4: By definition of exponent, $d\le q-1$
Therefore, $d=q-1$

Step 5: If the exponent equals the group order, the group is cyclic
There exists an element whose order is the exponent $q-1$

Conclusion: ${\mathbb{F}}^{\times }$ is cyclic.

109. Existence of finite field with pⁿ elements

Proof: Splitting field of x^{pⁿ}-x over ℤₚ has exactly pⁿ elements

110. Uniqueness of finite fields

Proof: Any two finite fields with same order are splitting fields of x^{q}-x, hence isomorphic

111. Sₙ not solvable for n≥5

Proof: Aₙ is simple non-Abelian for n≥5, so composition series has non-Abelian factor

112. Aₙ simple for n≥5

Proof: Technical proof showing no non-trivial proper normal subgroups

113. Fundamental Theorem of Algebra via Galois

Proof: No proper finite extension of ℂ, so every polynomial splits over ℂ

114. Prove that a polynomial is solvable by radicals if and only if its Galois group is solvable.

114. Prove that a polynomial is solvable by radicals if and only if its Galois group is solvable.

Proof Sketch:
This is the celebrated Galois Theorem.

(⇒) If polynomial solvable by radicals:

• Splitting field can be obtained by successive radical extensions

• Each radical extension corresponds to adjoining a root of $x^n-a$

• Galois group of radical extension is cyclic (or Abelian)

• Building tower of extensions gives solvable Galois group

(⇐) If Galois group solvable:

• Has composition series with Abelian factors

• Each factor corresponds to cyclic extension

• By Kummer theory, cyclic extensions are radical extensions

• Therefore, splitting field is radical extension of base field

Conclusion: The polynomial is solvable by radicals iff its Galois group is solvable.

Application: $S_5$ is not solvable ⇒ quintic equations not solvable by radicals.

115. Regular n-gon constructible iff φ(n) power of 2

Proof: Uses that constructible numbers lie in tower of quadratic extensions

116. Angle trisection impossible

Proof: 60° angle would require constructing 20° angle, but cos 20° has minimal polynomial of degree 3

117. Cube duplication impossible

Proof: Would require constructing ∛2, but minimal polynomial x³-2 has degree 3 not power of 2

118. π transcendental

Proof: Lindemann-Weierstrass theorem

119. e transcendental

Proof: Hermite's proof

120. Non-Abelian group of order 21

Example: Semidirect product ℤ₇ ⋊ ℤ₃

121. Non-commutative division ring

Example: Quaternions ℍ

122. PID not ED

Example: ℤ[(1+√-19)/2]

123. UFD not PID

Example: ℤ[x]

124. Integral domain not UFD

Example: ℤ[√-5]

125. Infinite-dimensional vector space

Example: ℝ[x] (polynomials over ℝ)VI. Mixed Application Problems

126. Show that the groups (Z₄, +) and (U₅, ×) are isomorphic.

126. Show that the groups (ℤ₄, +) and (U₅, ×) are isomorphic.

Solution:

• ${\mathbb{Z}}_4=\{0,1,2,3\}$ under addition mod 4

• $U_5=\{1,2,3,4\}$ under multiplication mod 5

Find isomorphism:
Both are cyclic of order 4:

• ${\mathbb{Z}}_4$ generated by 1

• $U_5$ generated by 2 (since $2^1=2,2^2=4,2^3=3,2^4=1$ mod 5)

Define φ: ${\mathbb{Z}}_4\to U_5$ by φ(k) = 2^k mod 5

Check isomorphism:

• φ(0) = 2^0 = 1

• φ(1) = 2^1 = 2

• φ(2) = 2^2 = 4

• φ(3) = 2^3 = 3

• Bijective

• φ(a+b) = 2^{a+b} = 2^a × 2^b = φ(a) × φ(b)

Answer: Yes, isomorphic via φ(k) = 2^k mod 5.

127. ℤ₂[x]/⟨x²+x+1⟩ ≅ ℤ₂[x]/⟨x²+1⟩?

No: x²+1=(x+1)² in ℤ₂[x], so not field

128. Determine all homomorphisms from Z₁₂ to Z₁₈.

128. Determine all homomorphisms from ${\mathbb{Z}}_{12}$ to ${\mathbb{Z}}_{18}$.

Solution:
Let $f:{\mathbb{Z}}_{12}\to {\mathbb{Z}}_{18}$ be a group homomorphism.

Key fact: A homomorphism from ${\mathbb{Z}}_n$ to any group is determined by the image of the generator 1.

Let $f(1)=a\in {\mathbb{Z}}_{18}$

Condition: The order of $a$ must divide 12 (since $12\cdot a=f(12)=f(0)=0$ in ${\mathbb{Z}}_{18}$)

In ${\mathbb{Z}}_{18}$, the possible orders of elements are divisors of 18: 1, 2, 3, 6, 9, 18

We need orders that divide 12: 1, 2, 3, 6

Find all elements in ${\mathbb{Z}}_{18}$ with these orders:

• Order 1: {0}

• Order 2: {9}

• Order 3: {6, 12}

• Order 6: {3, 15}

All homomorphisms:

1. $f(1)=0$ (trivial homomorphism)

2. $f(1)=9$

3. $f(1)=6$

4. $f(1)=12$

5. $f(1)=3$

6. $f(1)=15$

Verification: Each choice gives a well-defined homomorphism since $12a=0$ in ${\mathbb{Z}}_{18}$ for each $a$.

Answer: 6 homomorphisms total.

129. Automorphisms of S₃

Inn(S₃) ≅ S₃ (since center trivial)
No outer automorphisms, so Aut(S₃) ≅ S₃

130. Homomorphisms from (ℚ,+) to (ℚ,+)

For any r∈ℚ, f(x)=rx is homomorphism
These are all: if f(1)=r, then f(n)=nr, f(1/n)=r/n, so f(x)=rx

131. Every finite group isomorphic to permutation group

Cayley's Theorem: G ↪ S_{|G|} via left multiplication

132. V ≅ V for finite-dim V**

Natural isomorphism v ↦ (f ↦ f(v))

133. det: GL(n,F)→F homomorphism*

det(AB)=det(A)det(B)

134. ε: Sₙ→{±1} homomorphism

ε(στ)=ε(σ)ε(τ)

135. Center of Sₙ trivial for n≥3

Only identity commutes with all transpositions