Abstract Algebra

Basic (Undergraduate years 1–2)

Overview

Abstract Algebra is the branch of mathematics that studies algebraic structures. Moving away from concrete numbers and polynomials, it treats operations and their properties in an abstract, axiomatic manner. Developed throughout the 19th and 20th centuries, it forms a foundation of modern mathematics.

The central objects of abstract algebra are three algebraic structures: the group, the ring, and the field. These underpin essentially every area of mathematics, science, and engineering—the description of symmetry, the solution of equations, number theory, geometry, cryptography, and quantum mechanics, among others.

Hierarchy of Algebraic Structures

An algebraic structure is defined by equipping a set with one or two operations, and is characterized by the axioms that those operations must satisfy.

Hierarchy of algebraic structures (increasing structure)

$$\text{Set} \;\to\; \text{Group} \;\to\; \text{Abelian group} \;\to\; \text{Ring} \;\to\; \text{Commutative ring} \;\to\; \text{Integral domain} \;\to\; \text{Field}$$

The further right we go, the more operations and axioms are added, and the richer the structure becomes. Here the arrow $\to$ denotes the step of adding operations and axioms to strengthen the structure, not a literal set inclusion (a group has one operation, a ring has two, so the kind of object changes). Where the number of operations stays the same, genuine inclusions do hold: $\text{Abelian group}\subset\text{Group}$ and $\text{Field}\subset\text{Integral domain}\subset\text{Commutative ring}\subset\text{Ring}$. Moreover the additive part of a ring is an abelian group, and the nonzero elements of a field form an abelian group under multiplication, so each richer structure contains simpler ones inside it.

Figure 1. The nested hierarchy of algebraic structures. Each time we add an operation and axioms to the outer set, the structure becomes richer and narrows inward, $\text{Set}\to\text{Group}\to\text{Ring}\to\text{Field}$ (the nesting and arrows indicate increasing structure, not a literal set inclusion). Here $\mathbb{Z}$ is an integral domain (a ring that is not a field), while $\mathbb{Q},\mathbb{R},\mathbb{C}$ are fields in which division is possible.

Group

Definition: Group

A pair $(G, *)$ consisting of a set $G$ and a binary operation $*: G \times G \to G$ is a group if it satisfies the following axioms:

Associativity: $(a * b) * c = a * (b * c)$ for all $a, b, c \in G$
Existence of an identity: there exists $e \in G$ such that $e * a = a * e = a$ for all $a \in G$
Existence of inverses: for every $a \in G$ there exists $b \in G$ with $a * b = b * a = e$

If in addition $a * b = b * a$ for all $a, b \in G$, the group is called abelian (a commutative group).

Figure 2. A picture of a binary operation $*:G\times G\to G$. Feeding two elements $a,b$ of the set $G$ (the ellipse) into the operation $*$ (the circled symbol) produces an output $a*b$ that is again an element of $G$ (closure). The group axioms further constrain this $*$ with associativity, an identity, and inverses.

Examples

Addition of integers $(\mathbb{Z}, +)$: an abelian group (identity $0$, inverse is the sign change)
Multiplication of nonzero reals $(\mathbb{R}^*, \times)$: an abelian group (identity $1$, inverse is the reciprocal)
Symmetric group $S_n$: all permutations of $n$ elements (non-abelian for $n \geq 3$)
Rotation group $SO(3)$: all rotations of 3-dimensional space (non-abelian)
Residue classes $\mathbb{Z}/n\mathbb{Z}$: integers modulo $n$ (an abelian group)

Applications

As the mathematical expression of symmetry, groups are used in crystallography, molecular symmetry, particle physics (gauge theory), and cryptography (elliptic curve cryptography).

Ring

Definition: Ring

A triple $(R, +, \times)$ consisting of a set $R$ and two binary operations $+, \times$ is a ring if it satisfies the following:

$(R, +)$ is an abelian group (its identity is written $0$)
$\times$ is associative: $(ab)c = a(bc)$
Distributivity: $a(b + c) = ab + ac$, $(a + b)c = ac + bc$

Rings with the following additional properties are given special names:

Ring with identity (unital ring): a multiplicative identity $1$ exists
Commutative ring: $ab = ba$ for all $a, b \in R$
Integral domain: commutative, unital, and without zero divisors ($ab = 0 \Rightarrow a = 0$ or $b = 0$)

Examples

Ring of integers $\mathbb{Z}$: an integral domain (commutative, unital, no zero divisors)
Polynomial ring $\mathbb{R}[x]$: all polynomials with real coefficients (an integral domain)
Matrix ring $M_n(\mathbb{R})$: all $n \times n$ real matrices (non-commutative for $n \geq 2$)
Quotient ring $\mathbb{Z}/n\mathbb{Z}$: has zero divisors when $n$ is not prime

Applications

Ring theory is foundational to algebraic number theory, algebraic geometry, coding theory, and cryptography. Ideal theory (Noetherian rings, Dedekind rings) is important as a generalization of number theory.

Field

Definition: Field

A commutative ring $F$ is a field if every nonzero element has a multiplicative inverse. That is:

$(F, +)$ is an abelian group
$(F \setminus \{0\}, \times)$ is an abelian group
distributivity holds

Examples

Field of rational numbers $\mathbb{Q}$
Field of real numbers $\mathbb{R}$
Field of complex numbers $\mathbb{C}$
Finite field $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$ (where $p$ is prime)
Algebraic extension field: $\mathbb{Q}(\sqrt{2}) = \{a + b\sqrt{2} : a, b \in \mathbb{Q}\}$

Applications

Field theory plays a central role in the theory of equations (Galois theory), algebraic geometry, coding theory (Reed-Solomon codes), and cryptography (elliptic curves over finite fields).

Homomorphisms and Isomorphisms

Homomorphisms

A structure-preserving map between algebraic structures. A map $\phi: G \to H$ from a group $(G, *)$ to a group $(H, \cdot)$ is a homomorphism if $$\phi(a * b) = \phi(a) \cdot \phi(b) \quad \text{for all } a, b \in G$$ holds.

Here $*$ and $\cdot$ are the binary operations of the two groups, and we use distinct symbols to avoid confusion:

$*$ … the operation of the source group $G$
$\cdot$ … the operation of the target group $H$

In other words, the left-hand side $\phi(a * b)$ means “operate first in $G$, then send via $\phi$”, while the right-hand side $\phi(a) \cdot \phi(b)$ means “send via $\phi$ first, then operate in $H$”, and both give the same result (i.e. $\phi$ preserves the structure of the operation). This is the homomorphism condition.

Isomorphisms

A bijective homomorphism is called an isomorphism, written $G \cong H$. Isomorphic algebraic structures are regarded as “essentially the same”.

Example: $(\mathbb{R}, +) \cong (\mathbb{R}^+, \times)$ (the exponential function $\phi(x) = e^x$ is an isomorphism)

Major Theorems

Lagrange's Theorem

Lagrange's Theorem (group theory)

The order of a subgroup $H$ of a finite group $G$ divides the order of $G$: $$|H| \mid |G|$$

For a careful proof using the partition into cosets, see the dedicated article “Lagrange's Theorem”.

Homomorphism Theorem (First Isomorphism Theorem)

Homomorphism Theorem

For a group homomorphism $\phi: G \to H$, $$G / \ker(\phi) \cong \text{Im}(\phi)$$ (the quotient group by the kernel is isomorphic to the image).

For a complete proof covering the normality of $\ker\phi$, the well-definedness of the induced map, and injectivity and surjectivity, see the dedicated article “Group Homomorphisms and the Homomorphism Theorem”.

Chinese Remainder Theorem

Chinese Remainder Theorem (ring-theoretic version)

When $\gcd(m, n) = 1$, $$\mathbb{Z}/(mn)\mathbb{Z} \cong \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$$

For a constructive proof of the existence and uniqueness of solutions (down to the bijectivity as a ring isomorphism), see the dedicated number theory article “Chinese Remainder Theorem”.

Fundamental Theorems

Fundamental Theorem of Finitely Generated Abelian Groups: every finitely generated abelian group decomposes into a direct product of cyclic groups
Fundamental Theorem of Algebra: a degree-$n$ polynomial with complex coefficients has $n$ roots in the complex field (counted with multiplicity)
Fundamental Theorem of Galois Theory: the intermediate fields of a field extension correspond to the subgroups of the Galois group

Rigorous proofs of each fundamental theorem appear in the dedicated articles: the direct-sum decomposition into cyclic groups for the finite abelian case (existence plus uniqueness of the invariant factors) is in “Direct Products of Groups” (the finitely generated version reduces to the structure theorem for modules over a PID); the Fundamental Theorem of Algebra has an analytic proof using Liouville's theorem in “Fundamental Theorem of Algebra”; and the Fundamental Theorem of Galois Theory is in “The Fundamental Theorem of Galois Correspondence”.

Applications

Cryptography

Abstract algebra is foundational to RSA encryption (integer factorization), elliptic curve cryptography (the group structure of elliptic curves over finite fields), and AES (arithmetic in the finite field $\mathbb{F}_{2^8}$).

Coding Theory

It is used in linear codes (vector spaces over finite fields), Reed-Solomon codes (polynomial rings), and BCH codes (extensions of finite fields).

Crystallography

The symmetries of crystals are classified by space groups (group theory). There are 230 known space groups.

Particle Physics

Gauge theory (the Standard Model) is described by the Lie group $SU(3) \times SU(2) \times U(1)$.

Computer Science

It is applied in formal language theory (regular languages and monoids), complexity theory (the computational complexity of groups), and quantum computing (the unitary group).

Historical Development

Évariste Galois (1811–1832): linked the solvability of equations with group theory (Galois theory)
Richard Dedekind (1831–1916): introduced ideal theory and the concept of a ring
Emmy Noether (1882–1935): the axiomatic formulation of abstract algebra, Noetherian rings
Emil Artin (1898–1962): modern field theory, the reconstruction of Galois theory

In the 20th century, the unifying perspective of category theory was added, and abstract algebra became the common language of modern mathematics.

References

Dummit, D. S., & Foote, R. M. (2003). Abstract Algebra (3rd ed.). Wiley.
Artin, M. (2010). Algebra (2nd ed.). Pearson.
Herstein, I. N. (1996). Abstract Algebra (3rd ed.). Wiley.
Lang, S. (2002). Algebra (Revised 3rd ed., Graduate Texts in Mathematics 211). Springer.
Hungerford, T. W. (1980). Algebra (Graduate Texts in Mathematics 73). Springer.
Fraleigh, J. B., & Brand, N. (2021). A First Course in Abstract Algebra (8th ed.). Pearson.
Weisstein, E. W. "Abstract Algebra." From MathWorld—A Wolfram Web Resource.

Frequently Asked Questions

What are the main objects of study in abstract algebra?

Abstract algebra studies algebraic structures such as groups, rings, fields, modules, and vector spaces in an abstract, axiomatic way. The recurring themes are symmetry, operations, homomorphisms, isomorphisms, and invariants. Galois theory (the correspondence between field extensions and groups) is one of its most celebrated results.

How are groups, rings, and fields related by inclusion?

We have field $\subset$ commutative ring $\subset$ ring, and field $\subset$ division ring (skew field). A field is “an additive group and a multiplicative group (excluding zero)”, a commutative ring has “commutative multiplication and possibly zero divisors”, and a general ring has “possibly non-commutative multiplication and possibly zero divisors”. $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{F}_p$ are fields, while $\mathbb{Z}$ is an integral domain (a ring that is not a field).

How do you use the isomorphism theorems?

The standard procedure for proving that two algebraic structures are isomorphic: (1) construct a homomorphism $\phi: A \to B$, (2) identify the kernel $\ker\phi$, (3) apply the First Isomorphism Theorem $A/\ker\phi \cong \text{Im}\phi$, (4) check that $\text{Im}\phi = B$ (surjectivity). This yields $A/\ker\phi \cong B$.