Appendix C. Abstract Algebra Review

C.1 Algebraic Structures

Abstract algebra studies sets equipped with operations. In number theory, these structures organize arithmetic behavior.

The integers $\mathbb{Z}$ form a set with addition and multiplication. The residue classes modulo $n$ , written $\mathbb{Z}/n\mathbb{Z}$ , form a finite arithmetic system. Rational numbers, real numbers, complex numbers, finite fields, number fields, rings of integers, ideals, and elliptic curves all carry algebraic structure.

The purpose of abstraction is to isolate the laws that make arithmetic work. Once these laws are stated clearly, the same argument can apply in many settings.

C.2 Binary Operations

A binary operation on a set $S$ is a rule that assigns to each ordered pair $(a,b)\in S\times S$ an element of $S$ .

Examples:

a+b,\qquad ab

are binary operations on $\mathbb{Z}$ .

Subtraction is also a binary operation on $\mathbb{Z}$ , but division is not, because $a/b$ may fail to be an integer.

Closure is the requirement that the result remains inside the set. For instance, $\mathbb{Z}$ is closed under addition and multiplication, while $\mathbb{Z}\setminus\{0\}$ is not closed under subtraction.

C.3 Groups

A group is a set $G$ with a binary operation satisfying four axioms.

Associativity:

(a b)c=a(b c)

for all $a,b,c\in G$ .

Identity element:

e a=a e=a

for all $a\in G$ .

Inverses:

a a^{-1}=a^{-1}a=e

for every $a\in G$ .

Closure:

ab\in G

for all $a,b\in G$ .

If the operation also satisfies

ab=ba

for all $a,b\in G$ , then $G$ is abelian.

The integers $\mathbb{Z}$ form an abelian group under addition. The nonzero rational numbers $\mathbb{Q}^{\times}$ form an abelian group under multiplication.

C.4 Subgroups

A subset $H\subseteq G$ is a subgroup if it is itself a group under the operation inherited from $G$ .

A practical subgroup test is:

a,b\in H \implies ab^{-1}\in H.

For additive groups, this becomes:

a,b\in H \implies a-b\in H.

The subgroups of $\mathbb{Z}$ are exactly

n\mathbb{Z}=\{nk:k\in\mathbb{Z}\}

for integers $n\ge0$ .

This fact is one of the algebraic roots of divisibility theory.

C.5 Cyclic Groups

A group $G$ is cyclic if there exists $g\in G$ such that every element of $G$ is a power of $g$ . In additive notation, every element is a multiple of $g$ .

The group generated by $g$ is

\langle g\rangle=\{g^k:k\in\mathbb{Z}\}.

In additive notation:

\langle g\rangle=\{kg:k\in\mathbb{Z}\}.

The group $\mathbb{Z}$ is cyclic, generated by $1$ . The group $\mathbb{Z}/n\mathbb{Z}$ is cyclic, generated by the class of $1$ .

Cyclic groups are fundamental because many finite arithmetic systems contain cyclic subgroups.

C.6 Orders of Elements

The order of an element $g\in G$ is the smallest positive integer $m$ such that

g^m=e.

If no such $m$ exists, $g$ has infinite order.

In additive notation, the order of $g$ is the smallest positive integer $m$ such that

mg=0.

For example, in $\mathbb{Z}/n\mathbb{Z}$ , the class of $a$ has order

\frac{n}{\gcd(a,n)}.

This formula is used repeatedly in modular arithmetic and finite group arguments.

C.7 Homomorphisms

A group homomorphism is a function

\varphi:G\to H

such that

\varphi(ab)=\varphi(a)\varphi(b)

for all $a,b\in G$ .

Homomorphisms preserve algebraic structure. They send identities to identities and inverses to inverses.

The kernel of $\varphi$ is

\ker \varphi=\{g\in G:\varphi(g)=e_H\}.

The image is

\operatorname{im}\varphi=\{\varphi(g):g\in G\}.

Kernels measure which elements collapse to the identity. Images measure which elements are reached.

C.8 Rings

A ring is a set $R$ with two operations, addition and multiplication, satisfying these conditions:

The set $R$ is an abelian group under addition.

Multiplication is associative.

Multiplication distributes over addition:

a(b+c)=ab+ac,

(a+b)c=ac+bc.

Many texts also require a multiplicative identity $1$ . In number theory, rings usually have such an identity.

Examples include

\mathbb{Z},\quad \mathbb{Q},\quad \mathbb{R},\quad \mathbb{C},\quad \mathbb{Z}/n\mathbb{Z}.

Rings provide the natural setting for divisibility, congruences, ideals, and algebraic integers.

C.9 Integral Domains

A commutative ring with identity is an integral domain if

ab=0 \implies a=0 \text{ or } b=0.

Equivalently, it has no zero divisors.

The integers $\mathbb{Z}$ form an integral domain. The ring $\mathbb{Z}/n\mathbb{Z}$ is an integral domain exactly when $n$ is prime.

For example, in $\mathbb{Z}/6\mathbb{Z}$ ,

2\cdot3\equiv0\pmod6,

even though neither $2$ nor $3$ is congruent to $0$ modulo $6$ . Thus $\mathbb{Z}/6\mathbb{Z}$ has zero divisors.

C.10 Fields

A field is a commutative ring in which every nonzero element has a multiplicative inverse.

Examples:

\mathbb{Q},\quad \mathbb{R},\quad \mathbb{C}.

The finite ring $\mathbb{Z}/p\mathbb{Z}$ is a field exactly when $p$ is prime.

Fields are central in modern number theory. They support linear algebra, polynomial factorization, Galois theory, local analysis, and arithmetic geometry.

C.11 Units

A unit in a ring $R$ is an element $u\in R$ with a multiplicative inverse in $R$ . That is, there exists $v\in R$ such that

uv=vu=1.

The units of $\mathbb{Z}$ are

\pm1.

The units of $\mathbb{Z}/n\mathbb{Z}$ are the residue classes $a$ satisfying

\gcd(a,n)=1.

They form a group under multiplication, written

(\mathbb{Z}/n\mathbb{Z})^{\times}.

This group is the main object behind Euler’s theorem and many cryptographic constructions.

C.12 Ideals

An ideal $I$ of a commutative ring $R$ is an additive subgroup satisfying

r x\in I

for all $r\in R$ and $x\in I$ .

Ideals generalize divisibility. In $\mathbb{Z}$ , every ideal has the form

n\mathbb{Z}.

The ideal $n\mathbb{Z}$ consists of all multiples of $n$ .

In rings of algebraic integers, ideals restore unique factorization when elements themselves may fail to factor uniquely.

C.13 Principal Ideals

An ideal generated by a single element $a\in R$ is called principal and is written

(a)=\{ra:r\in R\}.

In $\mathbb{Z}$ ,

(a)=a\mathbb{Z}.

A ring in which every ideal is principal is called a principal ideal domain, or PID.

The ring $\mathbb{Z}$ is a PID. This fact underlies the Euclidean algorithm, greatest common divisors, and Bézout identities.

C.14 Quotient Rings

If $I$ is an ideal of $R$ , the quotient ring $R/I$ is the set of cosets

a+I=\{a+x:x\in I\}.

Addition and multiplication are defined by

(a+I)+(b+I)=(a+b)+I,

(a+I)(b+I)=ab+I.

The familiar ring of residues modulo $n$ is

\mathbb{Z}/n\mathbb{Z}.

It is the quotient of $\mathbb{Z}$ by the ideal $n\mathbb{Z}$ .

C.15 Polynomial Rings

Given a ring $R$ , the polynomial ring $R[x]$ consists of expressions

a_0+a_1x+\cdots+a_nx^n

with coefficients $a_i\in R$ .

Polynomial rings connect algebra and arithmetic. They appear in field extensions, minimal polynomials, algebraic integers, finite fields, and elliptic curves.

If $F$ is a field, then $F[x]$ has a Euclidean algorithm similar to the one in $\mathbb{Z}$ . This makes polynomial arithmetic over fields especially well behaved.

C.16 Irreducible and Prime Elements

In an integral domain, a nonzero nonunit element $r$ is irreducible if

r=ab

implies that $a$ or $b$ is a unit.

A nonzero nonunit element $p$ is prime if

p\mid ab \implies p\mid a \text{ or } p\mid b.

Every prime element is irreducible. In many familiar rings, such as $\mathbb{Z}$ , every irreducible element is prime. In more general rings this can fail.

This distinction is one reason algebraic number theory uses ideals.

C.17 Field Extensions

A field extension is an inclusion of fields

K\subseteq L.

We then view $L$ as a vector space over $K$ . Its dimension is called the degree of the extension and is written

[L:K].

For example,

\mathbb{Q}\subseteq\mathbb{Q}(\sqrt{2})

has degree $2$ , since every element has the form

a+b\sqrt{2}

with $a,b\in\mathbb{Q}$ .

Field extensions are the language of algebraic numbers and Galois theory.

C.18 Summary of Core Structures

Structure	Operations	Main Condition	Number-Theoretic Role
Group	One operation	Inverses exist	Symmetry, units, residue classes
Ring	Addition and multiplication	Distributive laws	Divisibility, congruences, ideals
Integral domain	Ring	No zero divisors	Factorization theory
Field	Ring	Nonzero elements invertible	Extensions, finite fields, algebraic methods
Ideal	Additive subgroup	Closed under multiplication by ring elements	Quotients and generalized divisibility
Module	Addition and scalar multiplication	Linear structure over a ring	Lattices, class groups, Galois modules

Abstract algebra supplies the structural language of modern number theory. Classical arithmetic begins with integers. Modern arithmetic studies the algebraic systems that integers generate.