PGTRB UNIT 1: ALGEBRA - COMPLETE CONCEPTS
Groups:
Binary Operator: Let be a non-empty set. A binary operation on satisfies the closure axiom: . The structure is called a groupoid.
Group: The structure is a group if it satisfies:
Closure:
Associativity:
Identity: such that
Inverse: such that
Abelian Group: A group is abelian if it is commutative: .
Remarks:
Semigroup: Closure + Associativity.
Monoid: Closure + Associativity + Identity.
Trivial Group: .
Finite/Infinite Group: Based on the number of elements.
Order of a Group: Denoted by , it is the number of distinct elements in a finite group.
Properties of a Group:
The identity element is unique.
The inverse of an element is unique.
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Cancellation Laws: and .
The equations and have unique solutions in .
For an abelian group, .
If , then is abelian.
If , then is abelian.
Permutation Groups:
A one-one mapping of a finite set onto itself is a permutation of degree .
Identity Permutation: .
Inverse Permutation: If , then .
Product/Composition: The product of two permutations and is denoted . In general, .
Symmetric Group : The group of all permutations of a set of elements.
Cyclic Permutation (Cycle): A permutation that cyclically permutes a subset of elements.
A cycle of length is an -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.
has even and odd permutations.
Alternating Group : The set of all even permutations of , which is a normal subgroup of index 2.
Cyclic Group:
A group is cyclic if such that every element is of the form for some integer . The element is called a generator of .
Remarks:
Every cyclic group is abelian, but the converse is not true.
If is a generator, then is also a generator.
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Every group of prime order is cyclic.
Every subgroup of a cyclic group is cyclic.
Every isomorphic image of a cyclic group is cyclic.
Cyclic groups of the same order are isomorphic.
If a finite group of order contains an element of order , it is cyclic.
The number of generators of a finite cyclic group of order is (Euler's totient function).
An infinite cyclic group has precisely 2 generators.
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 in : .
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 is an isomorphism if it is a bijective homomorphism.
Subgroup:
A non-empty subset of a group is a subgroup if 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 is a subgroup of , then .
If and are subgroups of an abelian group , then is a subgroup.
Euler's & Fermat's Theorems:
Euler's: If , then .
Fermat's: If is prime, then .
Normal Subgroups:
A subgroup of is normal if . Denoted .
Simple Group: A group with no proper normal subgroups.
The center of a group is a normal subgroup.
A subgroup of index 2 is normal.
Theorems:
iff .
The intersection of normal subgroups is normal.
If and is a subgroup of with , then .
If and is any subgroup of , then is a subgroup of and .
If is a cyclic normal subgroup of , then every subgroup of is normal in .
Conjugate Element:
is conjugate to if such that .
Normalizer:
is a subgroup of .
Quotient Group:
If , the set of cosets forms a group under coset multiplication.
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Every quotient group of a cyclic (or abelian) group is cyclic (or abelian).
Homomorphism:
A mapping is a homomorphism if .
Types: Isomorphism (bijective), Monomorphism (injective), Epimorphism (surjective).
Endomorphism: Homomorphism of a group into itself.
Automorphism: Isomorphism of a group onto itself.
Kernel: . The kernel is a normal subgroup of .
is an isomorphism iff .
Fundamental Theorem: Every homomorphic image of is isomorphic to a quotient group of (i.e., ).
Isomorphism Theorems:
First: Let be an epimorphism with kernel . If and , then .
Second: If and are normal subgroups of with , then .
Third: If is a subgroup and is a normal subgroup of , then .
Cauchy's Theorem (for Abelian Groups):
If a prime divides for a finite abelian group , then has an element of order .
Sylow Theorems:
Let , where is prime and . A subgroup of order is a Sylow -subgroup.
First: Sylow -subgroups exist.
Second: Any two Sylow -subgroups are conjugate.
Third: The number of Sylow -subgroups is of the form and divides .
If there is only one Sylow -subgroup, it is normal.
Center of a Group:
. It is a subgroup of .
Groups of order (p prime) are abelian.
Solvable and Nilpotent Groups:
Solvable Group: for some .
Nilpotent Group: for some .
Every nilpotent group is solvable.
A group of order is nilpotent.
For , is simple and is not solvable.
Rings:
An algebraic structure is a ring if:
is an abelian group.
is closed and associative.
Distributive laws hold.
Commutative Ring: Multiplication is commutative.
Ring with Unity: Contains a multiplicative identity (1).
Null Ring / Zero Ring: .
Boolean Ring: .
P-ring: .
Idempotent Element: .
Zero Divisor: Non-zero elements such that .
Cancellation Laws hold in a ring iff it has no zero divisors.
Integral Domain:
A commutative ring with unity and no zero divisors.
Examples: (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: .
Every field is an integral domain. Every finite integral domain is a field.
Subrings:
A non-empty subset of a ring is a subring if is itself a ring.
Test: and .
The intersection of subrings is a subring.
Subfield:
A non-empty subset of a field is a subfield if is a field.
Test: and (for ).
Characteristic of a Ring:
The smallest positive integer such that . If no such 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 .
Ideals:
A non-empty subset of a ring is an ideal if:
is a subgroup of .
and .
Left Ideal: .
Right Ideal: .
The intersection, sum, and product of two ideals are ideals.
Homomorphism of Rings:
A mapping is a homomorphism if and .
, .
is a subring of .
Kernel: . The kernel is an ideal of .
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 is an ideal of , then is a ring under coset operations.
Fundamental Theorem: If is an epimorphism, then .
Principal Ideal:
An ideal generated by a single element , denoted .
Principal Ideal Domain (PID): An integral domain where every ideal is principal.
Examples: , any field.
Prime Ideals:
An ideal in a commutative ring is prime if or .
In a commutative ring with unity, every maximal ideal is prime.
Maximal Ideals:
An ideal is maximal if no ideal lies strictly between and .
In a commutative ring with unity, an ideal is maximal iff is a field.
Quotient Field:
Every integral domain can be embedded in a field (its field of fractions). The quotient field of is .
Euclidean Domain:
An integral domain with a function such that:
for .
such that with or .
Examples: , 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 PID UFD.
Polynomial Rings:
, the set of all polynomials over a ring .
Degree: The highest power of with a non-zero coefficient.
If is an integral domain, then is an integral domain and .
If is a UFD, then is a UFD.
If is a field, then 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: is irreducible over . is irreducible over .
Vector Spaces:
A set over a field with vector addition and scalar multiplication satisfying:
is an abelian group.
Closure and compatibility conditions for scalar multiplication.
Examples: , , , .
Subspaces:
A non-empty subset of a vector space is a subspace if it is itself a vector space.
Test: ; .
The intersection of subspaces is a subspace.
Linear Combination & Linear Span:
Linear Combination: .
Linear Span : The set of all linear combinations of a subset . It is the smallest subspace containing .
Linear Dependence/Independence:
A set is linearly dependent if there exist scalars, not all zero, such that . 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.
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Linear Transformations:
A mapping between vector spaces over is linear if .
Kernel/Null Space: . It is a subspace of .
Range/Image: . It is a subspace of .
Rank-Nullity Theorem: .
Isomorphism of Vector Spaces:
A bijective linear transformation. Two finite-dimensional vector spaces are isomorphic iff they have the same dimension.
Quotient Space:
, where is a subspace of . .
Direct Sum:
if and .
Linear Functional & Dual Space:
Linear Functional: A linear transformation from to its field .
Dual Space : The vector space of all linear functionals on . .
Dual Basis: If is a basis for , then the dual basis for is defined by .
Annihilator: For a subset , . .
Minimal Polynomial:
The monic polynomial of least degree that annihilates a linear operator .
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 consisting of eigenvectors of .
Finite Fields (Galois Fields):
A field with a finite number of elements, , where is prime.
The multiplicative group of a finite field is cyclic.
Every element satisfies .
Field Extensions:
A field is an extension of if . Denoted .
Degree of Extension : The dimension of as a vector space over .
Finite Extension: is finite.
Tower Law: If , then .
Algebraic Elements:
An element is algebraic over if it is a root of some non-zero polynomial in .
Minimal Polynomial: The monic irreducible polynomial in of which is a root. Its degree is the degree of .
is a finite extension of with .
Algebraic Extension:
An extension where every element of is algebraic over . Every finite extension is algebraic.
Splitting Field:
The smallest field extension over which a polynomial 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 : The group of all -automorphisms of .
Fixed Field: The set of elements in fixed by every automorphism in a subgroup of .
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 and the subgroups of .
Inner Product Spaces:
A vector space with an inner product .
Norm: .
Schwarz's Inequality: .
Triangle Inequality: .
Orthogonal Vectors: .
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 of all real numbers forms an abelian group under .
Example 2: The sets (cube roots of unity) and (fourth roots of unity) form an abelian group under multiplication.
Example 3: The set forms an abelian group under .
Example 4: The set forms a group under .
Example 5: The set of matrices where and , forms an infinite abelian group under matrix multiplication.
Example 6: is a group. is an abelian group if is prime.
Example 7: The set of all bilinear transformations , where , , is an infinite abelian group.
Example 8: is a semigroup. is a monoid.
Example 9: is an abelian group. is a monoid.
Example 10: and are abelian groups. and are abelian groups.
2. Permutation & Symmetric Group Problems
Example: For , given permutations:
, .Find the product .
Find the product .
Example: Show that is a cycle of length 3.
Example: Show that is a cycle of length 5.
Example: Show that is not a cycle.
Example: Find the product of cycles .
Problem: Find the number of distinct 3-cycles in .
Solution: .
3. Cyclic Group Problems
Example 1: under multiplication is cyclic. Generators: .
Example 2: under multiplication is cyclic. Generators: .
Example 3: under addition modulo 6 is cyclic. Generators: .
Example 4: The set of roots of unity is cyclic. Generator: .
Problem: Find the number of generators of a finite cyclic group of order 28.
Solution: .
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 in a cyclic group of order is .
5. Ring and Field Examples
Examples of Rings: (non-commutative), .
Example of Boolean Ring: .
Example of Ring without unit: The set of all even integers.
Examples of Rings with zero divisors: The ring of all matrices over integers; .
Examples of Rings without zero divisors: .
Examples of Integral Domains: (p prime).
Examples of Fields: (p prime).
Example: The set of numbers of the form , , is a field.
Example: The Gaussian integers form a commutative ring with unity and an integral domain, but not a field.
Example: The set is not a ring.
6. Ideal Examples
Example: is a left ideal of the ring of all matrices.
Example: and are ideals in .
7. Polynomial Irreducibility Problems
Problem 1: Show that is irreducible over .
Problem 2: Show that is irreducible over .
Problem 3: Show that is irreducible over if is prime.
Problem 4: Show that is irreducible over .
Problem 5: Show that is reducible over since in .
Problem 6: Show that is reducible over since .
Problem 7: Show that is irreducible over .
8. Vector Space & Basis Problems
Problem 1: Determine if is linearly dependent. (Answer: Yes)
Problem 2: Determine if is linearly dependent. (Answer: Yes)
Problem 3: Determine if is linearly independent. (Answer: Yes)
Problem 4: Show that is linearly independent.
Problem 5: Show that form a basis of .
9. Minimal Polynomial Problems
Problem 1: Find the minimal polynomial of .
Solution: , so the minimal polynomial is .
Problem 2: Find the minimal polynomial of .
Solution: The characteristic polynomial is , which is also the minimal polynomial.
10. Field Extension & Splitting Field Problems
Problem 1: Show that is algebraic over of degree 2. (Minimal polynomial: )
Problem 2: Show that is algebraic over . (Minimal polynomial: )
Problem 3: Show that is algebraic over of degree 3. (Minimal polynomial: )
Problem 4: Find the splitting field and its degree over for .
Solution: . Splitting field: . Degree: .
Problem 5: Find the splitting field of over .
Solution: Splitting field: . Degree: 4.
Problem 6: Find the degree of the splitting field of over . (Answer: 6)
Problem 7: Find the splitting field of over .
Solution: . The roots are primitive 3rd and 6th roots of unity. Splitting field: where . Degree: 2.
Problem 8: Find the splitting field and degree of over .
Solution: Splitting field: . Degree: 8.
11. Inner Product Space & Gram-Schmidt Problem
Problem: Apply the Gram-Schmidt process to the basis of to find an orthonormal basis.
Solution Steps:
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Orthonormal Basis:
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 aβΏ = 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 aβΏ 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?
- Ο(n), where Ο 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 Sβ?
- 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 Aβ?
- The subgroup of Sβ 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 pα΅ * n, where p does not divide n, a Sylow p-subgroup is a subgroup of order pα΅.
33. State Sylow's First Theorem.
- For every prime power pα΅ dividing |G|, there exists a subgroup of order pα΅.
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 Sβ solvable for n > 4?
- No, Sβ 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) = 0β}. 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} → β₀ 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 Ξ±, Ξ² in W and a in F, (i) Ξ± - Ξ² ∈ W, and (ii) aΞ± ∈ W.
79. Define linear dependence of vectors.
- A set of vectors {v₁, v₂, ..., vβ} is linearly dependent if there exist scalars a₁, a₂, ..., aβ, not all zero, such that a₁v₁ + a₂v₂ + ... + aβvβ = 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(aΞ± + bΞ²) = aT(Ξ±) + bT(Ξ²) for all Ξ±, Ξ² in U and a, b in F.
86. Define the kernel (null space) of a linear transformation T.
- Ker(T) = {Ξ± in U | T(Ξ±) = 0}. It is a subspace of U.
87. Define the range (image) of a linear transformation T.
- R(T) = {T(Ξ±) | Ξ± 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(Ξ±) = 0 for all Ξ± 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 - Ξ»I).
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.
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(Ξ±) for some Ξ± in K.
101. Define an algebraic element.
- An element Ξ± in an extension K/F is algebraic over F if there exists a non-zero polynomial f(x) in F[x] such that f(Ξ±) = 0.
102. Define a transcendental element.
- An element that is not algebraic.
103. What is the minimal polynomial of an algebraic element Ξ±?
- The unique monic irreducible polynomial in F[x] of smallest degree that has Ξ± as a root.
104. If Ξ± is algebraic over F of degree n, what is [F(Ξ±) : F]?
- [F(Ξ±) : 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 pβΏ.
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 pβΏ?
- Exactly one.
Inner Product Spaces
115. Define an inner product space.
- A vector space V over a field F (where F is β or β) 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.
- |<Ξ±, Ξ²>| ≤ ||Ξ±|| ||Ξ²|| for all vectors Ξ±, Ξ² in an inner product space.
117. What are orthogonal vectors?
- Two vectors Ξ± and Ξ² are orthogonal if their inner product is zero: <Ξ±, Ξ²> = 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 ()
Subspaces: Check contains zero, closed under addition and scalar multiplication
Normal subgroups: Check for all
Ideals: Check and for all
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 be a group.
Suppose and are both identity elements in .
Then by definition of identity:
Since is identity:
Since is identity:
Therefore, , so .
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 be a group with identity . Suppose an element has two inverses and .
Then by definition of inverse:
Now consider:
(identity property)
(since )
(associativity)
(since )
(identity property)
Therefore, , so the inverse is unique.
We denote the unique inverse of as .
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 gα΅∈H
Claim: H = ⟨gα΅⟩
For any h∈H, h = gα΅ for some m
Write m = qk + r, 0 ≤ r < k
Then gΚ³ = gα΅*(gα΅)⁻α΅ ∈ H
By minimality of k, r = 0, so m = qk
Thus h = (gα΅)α΅ ∈ ⟨gα΅⟩
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 be a group homomorphism with identity elements and .
Let
Show K is a subgroup:
Closure: If , then , so
Identity: , so
Inverse: If , then , so
Show K is normal:
For any and :
So for all .
Hence, 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 Ο: G/K → G' by Ο(gK) = f(g)
Well-defined: If g₁K = g₂K, then g₁⁻¹g₂∈K, so f(g₁) = f(g₂)
Homomorphism: Ο(g₁Kg₂K) = Ο(g₁g₂K) = f(g₁g₂) = f(g₁)f(g₂) = Ο(g₁K)Ο(g₂K)
Injective: If Ο(gK) = e', then f(g) = e', so g∈K, so gK = K
Surjective: For any g'∈G', ∃g∈G with f(g) = g', so Ο(gK) = g'
10. Prove that every finite integral domain is a field.
10. Prove that every finite integral domain is a field.
Proof:
Let be a finite integral domain. We need to show every non-zero element has a multiplicative inverse.
Let where .
Fix a non-zero element . Consider the map defined by .
Show is injective:
If , then
Since and has no zero divisors,
Since is finite and is injective, it is also surjective.
In particular, there exists some such that
So every non-zero element has a multiplicative inverse .
Therefore, 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∈⟨aβ⟩ for some n, contradiction
Uniqueness: Suppose p₁⋯pβ = q₁⋯qβ with pα΅’,qβ±Ό irreducible
Since p₁ divides some qβ±Ό and both irreducible, p₁ ~ qβ±Ό
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,...,vβ} be basis for ker T
Extend to basis {v₁,...,vβ,vβ₊₁,...,vβ} of V
Claim: {T(vβ₊₁),...,T(vβ)} is basis for range T
Linear independence: If Ξ£cα΅’T(vα΅’)=0, then T(Ξ£cα΅’vα΅’)=0, so Ξ£cα΅’vα΅’∈ker T
Thus Ξ£cα΅’vα΅’ is linear combination of v₁,...,vβ, so all cα΅’=0
Spanning: For any w∈range T, w=T(v) with v=Ξ£aα΅’vα΅’
Then w = Ξ£aα΅’T(vα΅’) = Ξ£α΅’₌β₊₁βΏ aα΅’T(vα΅’)
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,...,vβ} for V, {w₁,...,wβ} for W
Define T: V → W by T(vα΅’) = wα΅’ 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(Ξ») = Ξ»² - tr(A)Ξ» + 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 Ξ±,Ξ² be algebraic over F, with minimal polynomials f,g
Then F(Ξ±,Ξ²) is finite extension of F (degree ≤ deg f · deg g)
Thus every element of F(Ξ±,Ξ²) is algebraic over F
In particular, Ξ±±Ξ², Ξ±Ξ², Ξ±/Ξ² (Ξ²≠0) are in F(Ξ±,Ξ²), 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^{qβΏ}-x over F where q=|F|)
And K/F is separable (every finite field is perfect)
Thus K/F is Galois
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: β€₂ only (prime order)
Order 3: β€₃ only (prime order)
Order 4: β€₄ or β€₂×β€₂ (Klein 4-group)
Order 5: β€₅ only (prime order)
26. Classify all groups of order 6 up to isomorphism.
Solution:
Let be a group of order 6.
Case 1: G is Abelian
By fundamental theorem of finite Abelian groups,
Case 2: G is non-Abelian
The only non-Abelian group of order 6 is (the symmetric group on 3 elements).
Classification:
(cyclic, Abelian)
(non-Abelian)
Verification:
has elements under addition mod 6
has 6 elements:
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 , which is prime.
By Lagrange's Theorem: The order of any element divides 7, so possible orders are 1 or 7.
If has an element of order 7: Then is cyclic, generated by that element.
Classification: (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
Elements: 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:
β€₈
β€₄×β€₂
β€₂×β€₂×β€₂
29. Classify all Abelian groups of order 12 up to isomorphism.
29. Abelian groups of order 12:
12 = 2²·3, so:
β€₁₂ ≅ β€₄×β€₃
β€₂×β€₂×β€₃ ≅ β€₂×β€₆
30. Classify all Abelian groups of order 16 up to isomorphism.
30. Abelian groups of order 16:
16 = 2⁴, partitions of 4:
β€₁₆ (4)
β€₈×β€₂ (3,1)
β€₄×β€₄ (2,2)
β€₄×β€₂×β€₂ (2,1,1)
β€₂×β€₂×β€₂×β€₂ (1,1,1,1)
31. Classify all rings with 2 elements up to isomorphism.
31. Rings with 2 elements:
Only β€₂ up to isomorphism
32. Classify all rings with 3 elements up to isomorphism.
32. Rings with 3 elements:
Only β€₃ up to isomorphism (field of prime order)
33. Classify all rings with 4 elements up to isomorphism.
33. Rings with 4 elements:
β€₄
β€₂×β€₂
π½₄ (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 exist iff is prime
The only field of order 4 is
Classification:
(the field with 4 elements)
Elements: where
Verification:
is not an integral domain since mod 4
Only works
35. Classify all fields of order 4, 8, 9 up to isomorphism.
35. Fields:
Order 4: π½₄ = β€₂[x]/⟨x²+x+1⟩
Order 8: π½₈ = β€₂[x]/⟨x³+x+1⟩
Order 9: π½₉ = β€₃[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: β€β only
Order p²: β€_{p²} or β€β×β€β
Order 2p: β€_{2p} or Dβ (dihedral)
39. Classify all subgroups of the symmetric group S₃.
39. Classify all subgroups of the symmetric group S₃.
Solution:
has order 6.
By Lagrange's Theorem: Possible subgroup orders are 1, 2, 3, 6.
List all subgroups:
Order 1: (trivial subgroup)
Order 2:
Order 3: (this is , the alternating group)
Order 6: 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 () 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 β€₁₂:
β€₁₂ ⊳ β€₆ ⊳ β€₃ ⊳ β€₁ ⊳ {0}
or β€₁₂ ⊳ β€₄ ⊳ β€₂ ⊳ β€₁ ⊳ {0}
42. Classify all maximal ideals of the ring Zβ for n = 6, 8, 12.
42. Maximal ideals of β€β:
For n=6: ⟨2⟩, ⟨3⟩
For n=8: ⟨2⟩
For n=12: ⟨2⟩, ⟨3⟩
43. Classify all prime ideals of the ring Zβ for n = 12, 18.
43. Prime ideals of β€β:
For n=12: ⟨2⟩, ⟨3⟩
For n=18: ⟨2⟩, ⟨3⟩
44. Classify all units in the ring Z₁₈.
44. Units in β€₁₈:
{1,5,7,11,13,17} (numbers coprime to 18)
45. Classify all nilpotent elements in the ring Z₁₂.
45. Nilpotent elements in β€₁₂:
{0,6} (since 6²=0 mod 12)
III. Computational Examples
46. Find the Galois group of over .
Solution:
Splitting field:
Splitting field isDegree of extension: since minimal polynomial of is
Galois group: has order 2
Elements:Identity:
Non-trivial:
Group structure:
Answer:
47. Find the Galois group of x² + 1 over β.
47. Find the Galois group of over .
Solution:
Splitting field:
Splitting field isDegree of extension: since minimal polynomial of is
Galois group: has order 2
Elements:Identity:
Complex conjugation:
Group structure:
Answer:
48. Find the Galois group of x³ - 2 over β.
48. Galois group of x³-2 over β
Splitting field: β(∛2,Ο) where Ο²+Ο+1=0
Degree: [β(∛2,Ο):β] = [β(∛2,Ο):β(∛2)]·[β(∛2):β] = 2·3 = 6
Galois group: S₃ (all permutations of the 3 roots ∛2, Ο∛2, Ο²∛2)
49. Find the Galois group of x⁴ - 2 over β.
49. Galois group of x⁴-2 over β
Splitting field: β(⁴√2,i)
Degree: [β(⁴√2,i):β] = [β(⁴√2,i):β(⁴√2)]·[β(⁴√2):β] = 2·4 = 8
Galois group: D₄ (dihedral group of order 8)
50. Find the splitting field and its degree over β for x⁴ + x² + 1.
50. Splitting field of x⁴+x²+1 over β
x⁴+x²+1 = (x²+x+1)(x²-x+1)
Roots: (-1±i√3)/2, (1±i√3)/2 = Ο, Ο², -Ο, -Ο² where Ο³=1
Splitting field: β(Ο) = β(√-3)
Degree: 2
51. Find the splitting field and its degree over β for x⁴ + 1.
51. Splitting field of x⁴+1 over β
Roots: ±(1±i)/√2 = ±ΞΆ₈, ±ΞΆ₈³ where ΞΆ₈ = e^{Οi/4}
Splitting field: β(ΞΆ₈) = β(i,√2)
Degree: 4
52. Minimal polynomial of √2+√3 over β
Let Ξ± = √2+√3
Ξ±² = 5+2√6
Ξ±²-5 = 2√6
(Ξ±²-5)² = 24
Ξ±⁴-10Ξ±²+1 = 0
Minimal polynomial: x⁴-10x²+1
53. Minimal polynomial of 1+i over β
Let Ξ± = 1+i
Ξ±-1 = i
(Ξ±-1)² = -1
Ξ±²-2Ξ±+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 can be written as a product of transpositions.
For the 5-cycle :
Method 1 (standard decomposition):
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: sends 1 → 5
Then fixes 5
Then fixes 5
Then fixes 5
So 1 → 5 ✓
Start with 2: fixes 2
Then fixes 2
Then fixes 2
Then sends 2 → 1
So 2 → 1 ✓
Continue checking... it works.
Method 2 (adjacent transpositions):
Answer:
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 β€₂₀
β€₂₀ = {0,1,...,19}
Elements of order 4: numbers coprime to 20 with order 4 = {5,15}
Count: Ο(4) = 2
61. Generators of β€₁₇
All non-zero elements: {1,2,...,16} (prime order group)
62. Characteristic polynomial of [[1,2],[3,4]]
det(Ξ»I-A) = det[[Ξ»-1,-2],[-3,Ξ»-4]] = (Ξ»-1)(Ξ»-4)-6 = Ξ»²-5Ξ»-2
63. Minimal polynomial of [[2,0,0],[0,2,0],[0,0,3]]
Eigenvalues: 2,2,3
Characteristic poly: (Ξ»-2)²(Ξ»-3)
Minimal poly: (Ξ»-2)(Ξ»-3) since matrix ≠ 2I
64. Find the rank and nullity of the linear transformation T: β³ → β³ defined by T(x,y,z) = (x+y, y+z, z+x).
64. Find the rank and nullity of T: defined by T(x,y,z) = (x+y, y+z, z+x)
Solution:
Matrix representation:
Standard basis:Matrix:
Row reduce:
Rank = number of pivots = 3
By Rank-Nullity Theorem:
Answer: Rank = 3, Nullity = 0
65. Kernel and range of T: β³→β², 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 = β² (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 fα΅’ such that fα΅’(vβ±Ό)=Ξ΄α΅’β±Ό
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 β€₂
β€₂[x]: x², x²+1=(x+1)², x²+x=x(x+1), x²+x+1 irreducible
Only x²+x+1
70. Irreducible cubics over β€₃
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
π½₈ = β€₂[x]/⟨x³+x+1⟩
Elements: a+bΞ±+cΞ±² where Ξ±³+Ξ±+1=0, a,b,c∈β€₂
72. Field with 9 elements
π½₉ = β€₃[x]/⟨x²+1⟩
Elements: a+bΞ± where Ξ±²=-1, a,b∈β€₃
73. Inverse of 1+2i in β€[i]
(1+2i)(1-2i) = 1+4 = 5
Inverse = (1-2i)/5 not in β€[i] ⇒ no inverse in β€[i]
74. Factor x³-2x²-x+2 over β€₆
Over β€: (x-1)(x+1)(x-2)
Over β€₆: same but check mod 6
75. Solve over β€₇: 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)
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:
Closure:
etc. All products are in S
Associativity: Inherited from complex multiplication
Identity: and for all
Inverses:
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 .
Check group axioms:
Closure: If and , then
Associativity: Matrix multiplication is associative
Identity: has , and for all
Inverses: If , then exists and
Answer: Yes, it forms a group ().
78. Even integers subgroup of (β€,+)?
Yes: closed under +, -, contains 0
79. Odd integers subgroup of (β€,+)?
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: (β,+)→(β*,×), f(x)=2Λ£ homomorphism?
Yes: f(x+y)=2^{x+y}=2Λ£2ΚΈ=f(x)f(y)
83. f: (β,+)→(β,+), 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. β€[i] integral domain?
Yes: subring of β which is field
87. β€[√-5] UFD?
No: 6=2·3=(1+√-5)(1-√-5) with 2,3,1±√-5 irreducible
88. ⟨x²+1⟩ maximal in β[x]?
Yes: β[x]/⟨x²+1⟩ ≅ β field
89. ⟨x²+1⟩ prime in β[x]?
Yes: x²+1 irreducible over β, so quotient is field
90. Verify whether the set W = {(x,y,z) ∈ β³ | x+y+z = 0} is a subspace of β³.
90. Verify whether W = {(x,y,z) ∈ β³ | x+y+z = 0} is a subspace of β³.
Solution:
Use subspace test:
Non-empty: since
Closed under addition:
Let ,
Then and
Sum of coordinates:
SoClosed under scalar multiplication:
Let and
Then
Sum of coordinates:
So
Answer: Yes, W is a subspace of β³.
91. Verify whether the set W = {(x,y,z) ∈ β³ | x²+y²+z² = 1} is a subspace of β³.
91. Verify whether the set W = {(x,y,z) ∈ β³ | x²+y²+z² = 1} is a subspace of β³.
Solution:
Use subspace test:
Contains zero vector? :
So
Since W doesn't contain the zero vector, it cannot be a subspace.
Additional checks (not needed but informative):
Not closed under addition: and , but their sum has
Not closed under scalar multiplication: , but has
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. β(√2) field extension of β?
Yes: β(√2) = {a+b√2 | a,b∈β} is field
97. β(Ο) simple extension of β?
Yes: β(Ο) is simple transcendental extension
98. Verify whether the polynomial x³ - 2 is irreducible over β.
98. Verify whether the polynomial x³ - 2 is irreducible over β.
Solution:
Use Eisenstein's Criterion:
Polynomial:
Choose prime :
All coefficients except leading are divisible by 2
Constant term is divisible by 2 but not by
Leading coefficient 1 is not divisible by 2
By Eisenstein's Criterion, is irreducible over β.
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 .
Answer: Yes, irreducible over β.S1W`2D
99. x⁴+4 irreducible over β?
No: x⁴+4 = (x²+2x+2)(x²-2x+2)
100. β(√2,√3) Galois over β?
Yes: splitting field of (x²-2)(x²-3), separable, normal
101. β(∛2) Galois over β?
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
Number of Sylow 5-subgroups:
and
Possible
If , unique Sylow 5-subgroup is normal ⇒ G not simple
So assumeNumber of Sylow 3-subgroups:
and
Possible
If , unique Sylow 3-subgroup is normal ⇒ G not simple
So assume orIf :
Let be a Sylow 3-subgroup
So
Action of G on Sylow 3-subgroups gives homomorphism
Kernel is normal subgroup
If kernel trivial, but , contradiction
So kernel non-trivial normal subgroup ⇒ G not simpleOther 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 be a finite field with elements. Then has elements.
Let be the exponent of (the LCM of the orders of all elements).
Step 1: Every element satisfies
This follows from Lagrange's Theorem and the definition of exponent.
Step 2: The polynomial has at most roots in
Since is a field, a polynomial of degree has at most roots.
Step 3: But all elements of are roots of
Therefore,
Step 4: By definition of exponent,
Therefore,
Step 5: If the exponent equals the group order, the group is cyclic
There exists an element whose order is the exponent
Conclusion: is cyclic.
109. Existence of finite field with pβΏ elements
Proof: Splitting field of x^{pβΏ}-x over β€β has exactly pβΏ elements
110. Uniqueness of finite fields
Proof: Any two finite fields with same order are splitting fields of x^{q}-x, hence isomorphic
111. Sβ not solvable for n≥5
Proof: Aβ is simple non-Abelian for n≥5, so composition series has non-Abelian factor
112. Aβ 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 β, so every polynomial splits over β
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
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: is not solvable ⇒ quintic equations not solvable by radicals.
115. Regular n-gon constructible iff Ο(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. Ο transcendental
Proof: Lindemann-Weierstrass theorem
119. e transcendental
Proof: Hermite's proof
120. Non-Abelian group of order 21
Example: Semidirect product β€₇ ⋊ β€₃
121. Non-commutative division ring
Example: Quaternions β
122. PID not ED
Example: β€[(1+√-19)/2]
123. UFD not PID
Example: β€[x]
124. Integral domain not UFD
Example: β€[√-5]
125. Infinite-dimensional vector space
Example: β[x] (polynomials over β)VI. Mixed Application Problems
126. Show that the groups (Z₄, +) and (U₅, ×) are isomorphic.
126. Show that the groups (β€₄, +) and (U₅, ×) are isomorphic.
Solution:
under addition mod 4
under multiplication mod 5
Find isomorphism:
Both are cyclic of order 4:
generated by 1
generated by 2 (since mod 5)
Define Ο: by Ο(k) = 2^k mod 5
Check isomorphism:
Ο(0) = 2^0 = 1
Ο(1) = 2^1 = 2
Ο(2) = 2^2 = 4
Ο(3) = 2^3 = 3
Bijective
Ο(a+b) = 2^{a+b} = 2^a × 2^b = Ο(a) × Ο(b)
Answer: Yes, isomorphic via Ο(k) = 2^k mod 5.
127. β€₂[x]/⟨x²+x+1⟩ ≅ β€₂[x]/⟨x²+1⟩?
No: x²+1=(x+1)² in β€₂[x], so not field
128. Determine all homomorphisms from Z₁₂ to Z₁₈.
128. Determine all homomorphisms from to .
Solution:
Let be a group homomorphism.
Key fact: A homomorphism from to any group is determined by the image of the generator 1.
Let
Condition: The order of must divide 12 (since in )
In , 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 with these orders:
Order 1: {0}
Order 2: {9}
Order 3: {6, 12}
Order 6: {3, 15}
All homomorphisms:
(trivial homomorphism)
Verification: Each choice gives a well-defined homomorphism since in for each .
Answer: 6 homomorphisms total.
129. Automorphisms of S₃
Inn(S₃) ≅ S₃ (since center trivial)
No outer automorphisms, so Aut(S₃) ≅ S₃
130. Homomorphisms from (β,+) to (β,+)
For any r∈β, 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. Ξ΅: Sβ→{±1} homomorphism
Ξ΅(ΟΟ)=Ξ΅(Ο)Ξ΅(Ο)
135. Center of Sβ trivial for n≥3
Only identity commutes with all transpositions
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