Finite field

Definition

A finite field $F$ is a field with finitely many elements.

Properties

A finite field has $p^n$ elements, for some prime $p$ and positive integer $n$ . The characteristic of $F$ is $p$ .

Proof (show/hide)

Let $p$ be the characteristic of $F$ . Clearly $p \ne 0$ so $p$ is prime.

Let $K = \{0, 1, ..., -1\} = \{\sum^k 1: k=1,...,p\}$ . This is a subfield, isomorphic to $F_p$ . Therefore as a vector space, $F$ is isomorphic to $(F_p)^n$ and so $F$ has $p^n$ elements.

A finite field cannot be algebraically closed: take the polynomial $f(x) = 1 + \prod_{\alpha \in F} (x-\alpha)$ - this is identically $1$ .

For each prime power $p^n$ , there is exactly one field $F_{p^n}$ of that order (up to isomorphism). (TODO proof)

Finite field

Definition

Properties

See also