Abstract Algebra: Comprehensive Study Guide

1. Introduction to Abstract Algebra
2. Group Theory
3. Ring Theory
4. Field Theory
5. Practice Problems and Quizzes

Introduction to Abstract Algebra

Abstract algebra is the study of algebraic structures such as groups, rings, fields, modules, vector spaces, and algebras. It grew out of earlier work in solving polynomial equations and number theory, and it provides a framework that can be applied to many areas of mathematics and other sciences.

While calculus and linear algebra deal with specific types of mathematical objects (like real numbers, complex numbers, or matrices), abstract algebra focuses on the underlying structures and the operations defined on them.

Key Structures in Abstract Algebra

The main algebraic structures studied in abstract algebra include:

Structure	Components	Examples
Group	Set with one binary operation satisfying closure, associativity, identity, and inverse properties	Integers under addition, non-zero real numbers under multiplication
Ring	Set with two binary operations (addition and multiplication) where it forms a group under addition and multiplication is associative and distributive	Integers, polynomials with real coefficients
Field	Ring where multiplication is commutative and every non-zero element has a multiplicative inverse	Rational numbers, real numbers, complex numbers
Vector Space	Group with scalar multiplication satisfying certain axioms	ℝ², ℝ³, space of continuous functions

Example: Different Views of the Same Structure

Consider the set of integers ℤ:

(ℤ, +) is a group: It has an identity (0), each element has an inverse (-n for n), and addition is associative.
(ℤ, +, ×) is a ring: It is a group under addition, and multiplication distributes over addition.
However, (ℤ, +, ×) is not a field because most integers don't have multiplicative inverses within ℤ.

By contrast, (ℚ, +, ×) is a field because every non-zero rational number has a multiplicative inverse that is also a rational number.

Historical Development

Abstract algebra has its roots in:

Ancient mathematics: Solving polynomial equations and number theory.
19th century: Work by Galois (group theory) and Dedekind (ring theory).
20th century: Formalization of abstract structures and application to various fields.

Abstract algebra provides a powerful framework for understanding mathematical structures. By focusing on the properties of operations rather than specific sets of numbers, it reveals deep connections between seemingly different mathematical objects.

Group Theory

Basic Definitions and Examples

Definition: Group

A group is a set G with a binary operation • that satisfies the following axioms:

Closure: For all a, b ∈ G, a • b ∈ G
Associativity: For all a, b, c ∈ G, (a • b) • c = a • (b • c)
Identity: There exists an element e ∈ G such that for all a ∈ G, e • a = a • e = a
Inverse: For each a ∈ G, there exists an element a⁻¹ ∈ G such that a • a⁻¹ = a⁻¹ • a = e

If the operation is also commutative (a • b = b • a for all a, b ∈ G), then the group is called abelian or commutative.

Example 1: (ℤ, +) - Integers under addition

Closure: The sum of two integers is an integer
Associativity: (a + b) + c = a + (b + c)
Identity: 0 is the identity element
Inverse: The inverse of a is -a
This is an abelian group since a + b = b + a

Example 2: (ℤ₅, +) - Integers modulo 5

The group consists of elements {0, 1, 2, 3, 4} with addition modulo 5.

+	0	1	2	3	4
0	0	1	2	3	4
1	1	2	3	4	0
2	2	3	4	0	1
3	3	4	0	1	2
4	4	0	1	2	3

Example 3: (S₃, ∘) - Permutations on 3 elements

This is a non-abelian group with 6 elements representing all possible permutations of 3 objects.

Elements: {e, (1 2), (1 3), (2 3), (1 2 3), (1 3 2)}

Where e is the identity permutation, (1 2) represents swapping positions 1 and 2, etc.

Subgroups and Cyclic Groups

Definition: Subgroup

A subset H of a group G is a subgroup if H itself forms a group under the same operation as G.

Subgroup Test

A non-empty subset H of a group G is a subgroup if and only if:

For all a, b ∈ H, a • b ∈ H (closure)
For all a ∈ H, a⁻¹ ∈ H (inverse)

Note: Associativity is inherited from G, and the identity element must be in H by the inverse property.

One-Step Subgroup Test

A non-empty subset H of a group G is a subgroup if and only if for all a, b ∈ H, a • b⁻¹ ∈ H.

(⇒) If H is a subgroup, then for any a, b ∈ H, we know b⁻¹ ∈ H by the inverse property, and a • b⁻¹ ∈ H by closure.

(⇐) Assume for all a, b ∈ H, a • b⁻¹ ∈ H.
- For a ∈ H, taking b = a gives a • a⁻¹ = e ∈ H, so H contains the identity.
- For a ∈ H, taking b = e gives a • e⁻¹ = a ∈ H, confirming closure under the operation.
- For a ∈ H, taking b = e gives e • a⁻¹ = a⁻¹ ∈ H, so inverses exist in H.
Therefore, H is a subgroup of G.

Definition: Cyclic Group

A group G is cyclic if there exists an element g ∈ G such that every element of G can be written as a power of g. We write G = ⟨g⟩, and g is called a generator of G.

Example: Cyclic Groups

1. (ℤ, +) is a cyclic group generated by 1 (or -1): ℤ = ⟨1⟩ = {..., -2, -1, 0, 1, 2, ...}

2. (ℤ₆, +) is a cyclic group generated by 1: ℤ₆ = ⟨1⟩ = {0, 1, 2, 3, 4, 5}

3. But (ℤ₆, +) is also generated by 5: ⟨5⟩ = {0, 5, 4, 3, 2, 1}

4. And ℤ₆ = ⟨1⟩ = ⟨5⟩, but ⟨2⟩ = {0, 2, 4} is a proper subgroup of ℤ₆

Cosets and Lagrange's Theorem

Definition: Left and Right Cosets

Let H be a subgroup of G and g ∈ G:

The left coset of H with respect to g is gH = {g • h : h ∈ H}
The right coset of H with respect to g is Hg = {h • g : h ∈ H}

Lagrange's Theorem

If G is a finite group and H is a subgroup of G, then the order of H divides the order of G. In other words, |G| = |H| × [G:H], where [G:H] is the number of distinct cosets of H in G (the index of H in G).

Example: Cosets in ℤ₆

Consider H = {0, 3} as a subgroup of G = ℤ₆.

The left cosets of H are:

0 + H = {0, 3}
1 + H = {1, 4}
2 + H = {2, 5}

Note that |G| = 6, |H| = 2, and [G:H] = 3, verifying Lagrange's Theorem: 6 = 2 × 3.

Normal Subgroups and Quotient Groups

Definition: Normal Subgroup

A subgroup N of a group G is normal if for every g ∈ G and n ∈ N, g • n • g⁻¹ ∈ N. Equivalently, N is normal if gN = Ng for all g ∈ G.

We denote a normal subgroup as N ◅ G.

Theorem: In an abelian group, every subgroup is normal.

Definition: Quotient Group

If N is a normal subgroup of G, we can form the quotient group G/N, which is the set of all cosets of N in G with the operation (aN) • (bN) = (a • b)N.

Example: Quotient Group ℤ/4ℤ

Consider G = (ℤ, +) and N = 4ℤ = {..., -8, -4, 0, 4, 8, ...}.

The quotient group ℤ/4ℤ consists of the cosets:

0 + 4ℤ = {..., -8, -4, 0, 4, 8, ...}
1 + 4ℤ = {..., -7, -3, 1, 5, 9, ...}
2 + 4ℤ = {..., -6, -2, 2, 6, 10, ...}
3 + 4ℤ = {..., -5, -1, 3, 7, 11, ...}

This quotient group is isomorphic to ℤ₄, the group of integers modulo 4.

Group Homomorphisms

Definition: Group Homomorphism

A function φ: G → H between groups (G, •) and (H, ∗) is a homomorphism if for all a, b ∈ G, φ(a • b) = φ(a) ∗ φ(b).

Important Concepts Related to Homomorphisms

Kernel: ker(φ) = {g ∈ G : φ(g) = e_H}
Image: im(φ) = {φ(g) : g ∈ G}
Isomorphism: A bijective homomorphism

First Isomorphism Theorem

If φ: G → H is a group homomorphism, then:

ker(φ) is a normal subgroup of G
im(φ) is a subgroup of H
G/ker(φ) ≅ im(φ)

Example: A Homomorphism from ℤ to ℤ₆

Define φ: ℤ → ℤ₆ by φ(n) = n mod 6.

This is a homomorphism because φ(m + n) = (m + n) mod 6 = (m mod 6 + n mod 6) mod 6 = φ(m) + φ(n).

The kernel is ker(φ) = 6ℤ = {..., -12, -6, 0, 6, 12, ...}

The image is im(φ) = ℤ₆ = {0, 1, 2, 3, 4, 5}

And ℤ/6ℤ ≅ ℤ₆, confirming the First Isomorphism Theorem.

Ring Theory

Definitions and Examples

Definition: Ring

A ring is a set R equipped with two binary operations, usually called addition (+) and multiplication (·), satisfying the following axioms:

(R, +) is an abelian group
Multiplication is associative: (a · b) · c = a · (b · c) for all a, b, c ∈ R
Multiplication distributes over addition:
- a · (b + c) = a · b + a · c for all a, b, c ∈ R (left distributivity)
- (b + c) · a = b · a + c · a for all a, b, c ∈ R (right distributivity)

Additional properties that a ring may have:

A ring R is commutative if a · b = b · a for all a, b ∈ R
A ring R has a unity or identity if there exists an element 1 ∈ R such that 1 · a = a · 1 = a for all a ∈ R

Example 1: The ring of integers (ℤ, +, ×)

(ℤ, +) is an abelian group with identity 0
Multiplication is associative and commutative
Multiplication distributes over addition
It has a multiplicative identity: 1

Example 2: The ring of 2×2 matrices with real entries (M₂(ℝ), +, ×)

(M₂(ℝ), +) is an abelian group
Matrix multiplication is associative but not commutative
Matrix multiplication distributes over addition
It has a multiplicative identity: the identity matrix

Example 3: The ring of polynomials with real coefficients (ℝ[x], +, ×)

(ℝ[x], +) is an abelian group
Polynomial multiplication is associative and commutative
It has a multiplicative identity: the constant polynomial 1

Ideals and Quotient Rings

Definition: Ideals

An ideal I of a ring R is a subset of R that satisfies:

(I, +) is a subgroup of (R, +)
For all r ∈ R and a ∈ I, both r · a ∈ I and a · r ∈ I

Special types of ideals:

A left ideal satisfies (1) and: for all r ∈ R and a ∈ I, r · a ∈ I
A right ideal satisfies (1) and: for all r ∈ R and a ∈ I, a · r ∈ I
A principal ideal generated by an element a ∈ R is denoted (a) and consists of all elements of the form r · a · s where r, s ∈ R

Definition: Quotient Ring

If I is an ideal of a ring R, the quotient ring R/I consists of the cosets r + I for r ∈ R, with operations:

(r + I) + (s + I) = (r + s) + I
(r + I) · (s + I) = (r · s) + I

Example: The quotient ring ℤ/6ℤ

The ideal 6ℤ consists of all multiples of 6: {..., -12, -6, 0, 6, 12, ...}

The quotient ring ℤ/6ℤ consists of 6 cosets: 0 + 6ℤ, 1 + 6ℤ, 2 + 6ℤ, 3 + 6ℤ, 4 + 6ℤ, 5 + 6ℤ

Addition and multiplication tables (showing representatives):

+	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	0
2	2	3	4	5	0	1
3	3	4	5	0	1	2
4	4	5	0	1	2	3
5	5	0	1	2	3	4

×	1	2	3	4	5
0	0	0	0	0	0
1	1	2	3	4	5
2	2	4	0	2	4
3	3	0	3	0	3
4	4	2	0	4	2
5	5	4	3	2	1

Notice that ℤ/6ℤ has zero divisors: 2 × 3 = 0 (mod 6), even though neither 2 nor 3 is 0 (mod 6).

Integral Domains

Definition: Zero Divisors

A zero divisor in a ring R is a non-zero element a ∈ R such that there exists a non-zero element b ∈ R with a · b = 0 or b · a = 0.

Definition: Integral Domain

An integral domain is a commutative ring with unity and no zero divisors.

Theorem: Every field is an integral domain.

Examples of Integral Domains

The ring of integers ℤ is an integral domain
The ring of polynomials ℝ[x] is an integral domain
The ring ℤ₅ (integers modulo 5) is an integral domain

Examples that are not Integral Domains

The ring ℤ₆ is not an integral domain because 2 × 3 = 0 (mod 6)
The ring M₂(ℝ) of 2×2 matrices is not an integral domain (it's not even commutative)

Polynomial Rings

Definition: Polynomial Ring

Given a ring R, the polynomial ring R[x] consists of polynomials with coefficients from R:

a₀ + a₁x + a₂x² + ... + aₙxⁿ where a₀, a₁, ..., aₙ ∈ R

Theorem: If F is a field, then F[x] is a principal ideal domain.

Example: Polynomial Division Algorithm

Let F be a field. For any f(x), g(x) ∈ F[x] with g(x) ≠ 0, there exist unique q(x), r(x) ∈ F[x] such that:

f(x) = g(x)q(x) + r(x)

where either r(x) = 0 or deg(r(x)) < deg(g(x)).

Example Problem: Dividing Polynomials

Divide f(x) = x³ + 2x² - x + 3 by g(x) = x - 2

We use the polynomial long division method:

            x² + 4x + 7
   _____________________
x-2 ) x³ + 2x² - x + 3
       x³ - 2x²
       ________
          4x² - x
          4x² - 8x
          _______
               7x + 3
               7x - 14
               _______
                    17

Therefore, x³ + 2x² - x + 3 = (x - 2)(x² + 4x + 7) + 17

Our quotient is q(x) = x² + 4x + 7 and remainder is r(x) = 17

Field Theory

Definition and Examples

Definition: Field

A field is a commutative ring with unity where every non-zero element has a multiplicative inverse.

In other words, a field F satisfies:

(F, +) is an abelian group with identity 0
(F - {0}, ·) is an abelian group with identity 1
Multiplication distributes over addition: a · (b + c) = a · b + a · c for all a, b, c ∈ F

Examples of Fields

The rational numbers ℚ
The real numbers ℝ
The complex numbers ℂ
The finite field ℤ₅ (integers modulo 5)

Example: The Finite Field ℤ₅

Elements: {0, 1, 2, 3, 4}

Addition and multiplication tables:

+	0	1	2	3	4
0	0	1	2	3	4
1	1	2	3	4	0
2	2	3	4	0	1
3	3	4	0	1	2
4	4	0	1	2	3

×	1	2	3	4
0	0	0	0	0
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

Every non-zero element has a multiplicative inverse:

1⁻¹ = 1 (since 1 × 1 = 1)
2⁻¹ = 3 (since 2 × 3 = 6 ≡ 1 mod 5)
3⁻¹ = 2 (since 3 × 2 = 6 ≡ 1 mod 5)
4⁻¹ = 4 (since 4 × 4 = 16 ≡ 1 mod 5)

Field Extensions

Definition: Field Extension

A field extension E/F is a field E that contains F as a subfield.

We say that E is an extension of F, or F is a subfield of E.

Definition: Algebraic and Transcendental Elements

An element α ∈ E is algebraic over F if there exists a non-zero polynomial f(x) ∈ F[x] such that f(α) = 0.

If α is not algebraic over F, it is transcendental over F.

Definition: Degree of an Extension

The degree of the extension E/F, denoted [E:F], is the dimension of E as a vector space over F.

Theorem: Tower Law

If F ⊆ K ⊆ E are fields, then [E:F] = [E:K] × [K:F].

Example: Extension ℚ(√2)/ℚ

The field ℚ(√2) consists of all numbers of the form a + b√2 where a, b ∈ ℚ.

We can verify that √2 is algebraic over ℚ because it satisfies the polynomial x² - 2 = 0.

The degree of this extension is [ℚ(√2):ℚ] = 2, because {1, √2} forms a basis for ℚ(√2) as a vector space over ℚ.

Introduction to Galois Theory

Definition: Automorphism

An automorphism of a field F is an isomorphism from F to itself.

The set of all automorphisms of F forms a group under composition, denoted Aut(F).

Definition: Field Automorphism that Fixes a Subfield

If E/F is a field extension, an automorphism σ of E fixes F if σ(a) = a for all a ∈ F.

The set of all automorphisms of E that fix F forms a group, the Galois group of E over F, denoted Gal(E/F).

Fundamental Theorem of Galois Theory (Informal Statement)

If E/F is a finite Galois extension, there is a one-to-one correspondence between:

Intermediate fields K (where F ⊆ K ⊆ E)
Subgroups H of the Galois group Gal(E/F)

This correspondence reverses inclusion: larger fields correspond to smaller subgroups.

Example: Galois Group of ℚ(√2)/ℚ

Consider the field extension ℚ(√2)/ℚ.

Any automorphism σ of ℚ(√2) that fixes ℚ must satisfy:

σ(a) = a for all a ∈ ℚ
σ(√2)² = σ(2) = 2, so σ(√2) = ±√2

This gives us two possibilities:

The identity automorphism: id(a + b√2) = a + b√2
The conjugation automorphism: σ(a + b√2) = a - b√2

So Gal(ℚ(√2)/ℚ) = {id, σ} ≅ ℤ₂

Practice Problems and Quizzes

Ring Theory Quiz

4. Which of the following is an integral domain but not a field?

ℤ (integers) ℚ (rational numbers) ℤ₂ × ℤ₂ (direct product) ℤ₅ (integers modulo 5)

5. In the polynomial ring ℝ[x], what is the degree of the polynomial (x² + 1)(x³ - 2x + 4)?

3 5 6 8

Field Theory Quiz

6. Which of the following is NOT a field?

ℝ (real numbers) ℤ (integers) ℤ₇ (integers modulo 7) ℚ(√2) (rational numbers with √2 adjoined)

7. If [E:F] = 6 and [K:F] = 2 where F ⊆ K ⊆ E are fields, then [E:K] = ?

2 3 4 12

Worked Example: Group Theory Problem

Problem: Prove that a group G of order 15 must be cyclic.

By Lagrange's Theorem, the possible orders of subgroups of G are the divisors of 15: 1, 3, 5, and 15.

Let's analyze the structure of G:

By Cauchy's Theorem, G contains elements of order 3 and order 5.
Let a be an element of order 3 and b be an element of order 5.
Consider the subgroup H = ⟨a⟩ and K = ⟨b⟩. We have |H| = 3 and |K| = 5.
Since gcd(3, 5) = 1, H ∩ K = {e}.
For any h ∈ H and k ∈ K, we need to analyze the relationship between hk and kh.
Since |H| and |K| are coprime, one can show that H and K must commute, i.e., hk = kh for all h ∈ H, k ∈ K.
Therefore, HK = {hk : h ∈ H, k ∈ K} is a subgroup of G with |HK| = |H| × |K| / |H ∩ K| = 3 × 5 / 1 = 15.
This means G = HK = ⟨a⟩⟨b⟩ = ⟨ab⟩, i.e., G is cyclic.

Therefore, any group of order 15 must be cyclic.