### Table of Contents

## Fields

Abbreviation: **Fld**

### Definition

A \emph{field} is a commutative rings with identity $\mathbf{F}=\langle F,+,-,0,\cdot,1 \rangle$ such that

$\mathbf{F}$ is non-trivial: $0\ne 1$

every non-zero element has a multiplicative inverse: $x\ne 0\Longrightarrow \exists y (x\cdot y=1)$

Remark: The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.

##### Morphisms

Let $\mathbf{F}$ and $\mathbf{G}$ be fields. A morphism from $\mathbf{F}$ to $\mathbf{G}$ is a function $h:F\rightarrow G$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

Remark: It follows that $h(0)=0$ and $h(-x)=-h(x)$.

### Examples

Example 1: $\langle\mathbb{Q},+,-,0,\cdot,1\rangle$, the field of rational numbers with addition, subtraction, zero, multiplication, and one.

### Basic results

$0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$.

### Properties

### Finite members

$\begin{array}{lr}
f(1)= &0

f(2)= &1

f(3)= &1

f(4)= &1

f(5)= &1

f(6)= &0

\end{array}$

There exists one field, called the Galois field $GF(p^m)$ of each prime-power order $p^m$.