Introduction to Abstract Algebra

Overview

A GROUP is a set with one operation with nice properties.
A RING is a set with two operations (addition and multiplication). A ring is a group under addition.
A FIELD is a group under both addition and multiplication.

A group is a set $\mathcal{G}$ with an operation $\bullet$ satisfying:

1. Closure ( $\mathcal{G}$ is closed under $\bullet$ )

\forall a, b \in \mathcal{G} \; a \bullet b \in \mathcal{G}

2. Identity

\exists e \in \mathcal{G} \; \mathrm{s.t.} \; \forall a \in \mathcal{G} \; e \bullet a = a \bullet e = a

3. Inverse ( $e$ is the identity)

\forall a \in \mathcal{G} \; \exists b \in \mathcal{G} \; \mathrm{s.t.} \; a \bullet b = b \bullet a = e

4. Associativity

\forall a, b, c \in \mathcal{G} \; a \bullet (b \bullet c) = (a \bullet b) \bullet c

Abelian Group: A group where $\forall a, b \in G \; a \bullet b = b \bullet a$

$\mathbb{Z}/n\mathbb{Z}$ — integers mod n under modulo addition
$(\mathbb{Z}/n\mathbb{Z})^{\times}$ — integers mod n relatively prime to n under multiplication
$\mathbb{Z}$ — integers under addition (groups don’t have to be finite)

A ring is a set $\mathcal{R}$ satisfying:

1. Closure under $+$ and $\times$

\forall a, b \in \mathcal{R} \; a + b \in \mathcal{R} \;\textrm{and}\; a \times b \in \mathcal{R}

2. Abelian group under $+$

3. Associativity of $\times$

\forall a, b, c \in \mathcal{R} \; a \times (b \times c) = (a \times b) \times c

4. Distributive properties

\forall a, b, c \in \mathcal{R} \; a \times (b + c) = (a \times b) + (a \times c)

\forall a, b, c \in \mathcal{R} \; (b + c) \times a = (b \times a) + (c \times a)

A field is a set $\mathcal{F}$ satisfying:

1. Closure under $+$ and $\times$

2. Abelian group under $+$

3. $\mathcal{F} - \{e\}$ is an abelian group under $\times$ (where $e$ is the additive identity)

For prime $p$ , $\mathbb{Z}/p\mathbb{Z}$ (also written $\mathbb{F}_{p}$ ) under modulo operations
$\mathbb{Q}$ , $\mathbb{R}$ , $\mathbb{C}$ — infinite fields under regular operations

$\mathbb{Z}$ is NOT a field (no multiplicative inverses for non-unit integers)
$\mathbb{Z}/n\mathbb{Z}$ for composite $n$ is not a field (elements not relatively prime to $n$ have no inverse)