Field

Definition

A field $(F,+,\times)$ is a set $F$ with elements $0$ such that $(F,+)$ and $(F\setminus\{0\},\times)$ are abelian groups, and the distributivity law holds: for all $\forall a,b,c \in F: \; a(b+c) = ab + ac$ .

Equivalently, $F$ is a commutative ring whose units are $F^\times=F\setminus\{0\}$ .

The multiplicative inverse of $a$ is denoted $a^{-1}$ , and we often write $ab^{-1} = \frac{a}{b}$ .

Properties

As with all rings, a field has an additive identity $0$ and a multiplicative identity $1$ . In a field these are always distinct.

If $F, E$ are fields with the same operations and $F \subseteq E$ , we say $F$ is a subfield of $E$ and $E$ is a field extension of $F$

The characteristic of a field is the smallest positive integer $c \in \mathbb{N}$ such that $\sum^c x = 0$ for all $x \in F$ . If no such number exists, the characteristic is zero. The characteristic is always zero or prime.

Any finite field has size $p^n$ where $p$ is prime. Any finite field is not algebraically closed.

"Fields are the most boring kind of ring"

All non-zero elements are units, hence there are only two ideals: $F$ and $\{0\}$ . Therefore:

a field is a principal ideal domain.
the only quotient rings of a field are the ring itself and the zero ring.

Therefore by the first isomorphism theorem, any homomorphism from a field to an integral domain (or field) is either trivial or injective.

In fact, a field is a euclidean ideal domain, with norm $N(\alpha) = 1$ for all $\alpha \in F$ .
TODO stuff about homomorphisms. they sound like they might be boring...

Types

TODO algebraically closed

Examples

Any finite integral domain is a field.

The rational numbers $\mathbb{Q}$ , real numbers $\mathbb{R}$ and complex numbers $\mathbb{C}$ .

A commutative ring's field of fractions