Abstract Algebra

Introduction

Abstract algebra studies algebraic structures such as groups, rings, and fields. It generalises familiar number systems and provides powerful tools for understanding symmetry, solving equations, and modern cryptography.

The subject emerged in the 19th century from attempts to solve polynomial equations and understand geometric symmetries. Today it forms a cornerstone of modern mathematics, with applications ranging from cryptography to physics.

Groups

Definition

A group (G, •) is a set G together with a binary operation • satisfying four axioms:

1. Closure: For all a, b ∈ G, we have a • b ∈ G
2. Associativity: For all a, b, c ∈ G, we have (a • b) • c = a • (b • c)
3. Identity: There exists e ∈ G such that e • a = a • e = a for all a ∈ G
4. Inverses: For each a ∈ G, there exists a⁻¹ ∈ G such that a • a⁻¹ = a⁻¹ • a = e

If additionally a • b = b • a for all elements, the group is called abelian (or commutative).

Examples

• Integers under addition (ℤ, +) — identity is 0, inverse of n is −n
• Non-zero rationals under multiplication (ℚ*, ×) — identity is 1, inverse of q is 1/q
• Symmetric group Sₙ — all permutations of n elements, non-abelian for n ≥ 3
• Cyclic groups ℤₙ — integers mod n under addition, generated by a single element
• Dihedral groups Dₙ — symmetries of a regular n-gon (rotations and reflections)
• General linear group GL(n, ℝ) — invertible n×n matrices

Subgroups

A subgroup H of G is a subset that forms a group under the same operation.

Subgroup criterion: A non-empty subset H ⊆ G is a subgroup if and only if for all a, b ∈ H, we have ab⁻¹ ∈ H.

Lagrange’s theorem: If H is a subgroup of a finite group G, then |H| divides |G|. The quotient |G|/|H| is called the index of H in G.

A subgroup N is normal if gNg⁻¹ = N for all g ∈ G. Normal subgroups are precisely the kernels of homomorphisms.

Group Homomorphisms

A homomorphism φ: G → H is a function preserving the group operation:
$\varphi (a\cdot b)=\varphi (a)\cdot \varphi (b)$

Key concepts:

• Kernel: ker(φ) = {g ∈ G : φ(g) = eₕ} — always a normal subgroup
• Image: im(φ) = {φ(g) : g ∈ G} — always a subgroup of H
• Isomorphism: A bijective homomorphism; groups are “structurally identical”

First Isomorphism Theorem: If φ: G → H is a homomorphism, then G/ker(φ) ≅ im(φ).

Important Concepts

Order of an element: The smallest positive integer n such that gⁿ = e. If no such n exists, g has infinite order.

Cyclic groups: Generated by a single element g, written ⟨g⟩ = {gⁿ : n ∈ ℤ}. Every cyclic group is abelian.

Cosets: For subgroup H ⊆ G, the left coset of a is aH = {ah : h ∈ H}. Cosets partition G into equal-sized pieces.

Quotient groups: If N is normal in G, then G/N = {gN : g ∈ G} forms a group under coset multiplication.

Rings

Definition

A ring (R, +, ×) is a set R with two binary operations satisfying:

1. (R, +) is an abelian group
2. Multiplication is associative: (ab)c = a(bc)
3. Distributive laws hold: a(b + c) = ab + ac and (a + b)c = ac + bc

Types of Rings

• Commutative ring: Multiplication is commutative (ab = ba)
• Ring with unity: Has a multiplicative identity element 1
• Integral domain: Commutative ring with unity and no zero divisors (ab = 0 implies a = 0 or b = 0)
• Division ring: Every non-zero element has a multiplicative inverse

Examples

• Integers ℤ — the archetypal integral domain
• Polynomial rings R[x] — polynomials with coefficients from ring R
• Matrix rings Mₙ(R) — n×n matrices over R, non-commutative for n ≥ 2
• ℤₙ — integers mod n; an integral domain iff n is prime
• Gaussian integers ℤ[i] — complex numbers a + bi where a, b ∈ ℤ

Ideals

An ideal I of a ring R is a subring such that for all r ∈ R and a ∈ I, both ra ∈ I and ar ∈ I.

Principal ideal: Generated by a single element, (a) = {ra : r ∈ R}.

Prime ideal: If ab ∈ P implies a ∈ P or b ∈ P.

Maximal ideal: Proper ideal not contained in any larger proper ideal.

Quotient rings: R/I is a ring with operations [a] + [b] = [a + b] and [a][b] = [ab].

Key results:

• R/I is a field if and only if I is maximal
• R/I is an integral domain if and only if I is prime

Ring Homomorphisms

A ring homomorphism φ: R → S preserves both operations:

• φ(a + b) = φ(a) + φ(b)
• φ(ab) = φ(a)φ(b)

The kernel is always an ideal, leading to an analogue of the first isomorphism theorem.

Fields

Definition

A field is a commutative ring with unity where every non-zero element has a multiplicative inverse. Equivalently, a field has exactly two ideals: {0} and the ring itself.

Examples

• Rational numbers ℚ — smallest field containing ℤ
• Real numbers ℝ — complete ordered field
• Complex numbers ℂ — algebraically closed field
• Finite fields 𝔽ₚ — integers mod p for prime p
• Finite fields 𝔽ₚₙ — unique field of order pⁿ for each prime power

Characteristic

The characteristic of a field F is the smallest positive integer n such that n · 1 = 0, or 0 if no such n exists. The characteristic is always 0 or prime.

Field Extensions

A field extension L/K is a field L containing K as a subfield. L becomes a vector space over K.

• Degree: [L : K] = dim_K(L), the dimension as a K-vector space
• Algebraic extension: Every element satisfies a polynomial over K
• Transcendental extension: Contains elements not algebraic over K
• Splitting field: Smallest extension where a polynomial factors completely

Tower law: If M/L and L/K are extensions, then [M : K] = [M : L][L : K].

Galois Theory

Overview

Galois theory establishes a profound connection between field extensions and group theory, enabling us to understand when polynomial equations can be solved by radicals.

For a polynomial f(x) over field K, the Galois group Gal(L/K) consists of all automorphisms of the splitting field L that fix K.

Fundamental Theorem

There is a one-to-one correspondence between:

• Intermediate fields K ⊆ M ⊆ L
• Subgroups H of Gal(L/K)

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

Key Results

Solvability by radicals: A polynomial is solvable by radicals if and only if its Galois group is a solvable group.

Classical impossibilities (proved using Galois theory):

• Trisecting an arbitrary angle with compass and straightedge
• Squaring the circle (also requires transcendence of π)
• Doubling the cube
• General quintic formula — no formula exists for degree ≥ 5

Applications

Cryptography

• RSA encryption: Based on the difficulty of factoring in ℤ
• Elliptic curve cryptography: Groups of points on elliptic curves over finite fields
• Diffie-Hellman key exchange: Relies on discrete logarithm problem in cyclic groups

Coding Theory

• Error-correcting codes: Linear codes use vector spaces over finite fields
• Reed-Solomon codes: Polynomial evaluation over finite fields
• BCH codes: Based on properties of finite field extensions

Physics

• Symmetry groups: Describe fundamental symmetries (Lorentz group, SU(3), etc.)
• Quantum mechanics: Representations of groups classify particle states
• Crystallography: Space groups classify crystal structures

Chemistry

• Molecular symmetry: Point groups determine spectroscopic properties
• Orbital theory: Group representations predict chemical bonding

Glossary

Term	Definition
Abelian	A group where the operation is commutative
Kernel	Elements of a homomorphism mapping to the identity
Isomorphism	A bijective structure-preserving map
Ideal	A subring closed under multiplication by any ring element
Characteristic	Smallest n where n · 1 = 0, or 0 if none exists
Coset	A translate aH of a subgroup H
Normal subgroup	Subgroup invariant under conjugation
Simple group	Group with no proper normal subgroups
Solvable group	Has a composition series with abelian factors

Learning Resources

Books

• Abstract Algebra by Dummit and Foote — comprehensive graduate text
• A First Course in Abstract Algebra by Fraleigh — excellent introduction
• Algebra by Michael Artin — geometric intuition
• Topics in Algebra by Herstein — classic problems
• Galois Theory by Ian Stewart — accessible introduction to Galois theory

Online Courses

• Harvard Abstract Algebra — Benedict Gross lectures
• Socratica Abstract Algebra — YouTube series with clear explanations
• MIT OpenCourseWare 18.703 — Modern Algebra

Prerequisites

A solid understanding of Mathematical Proofs and basic Linear Algebra is essential before studying abstract algebra.

Related Topics

• Linear Algebra — vector spaces are modules over fields
• Number Theory — many results rely on ring and field theory
• Mathematical Proofs — abstract algebra requires rigorous proof techniques