Finite Fields

Order p^n; F^× cyclic of order p^n - 1.

The idea

Theorem (Classification of finite fields).

If a field has finitely many elements, then it has $p^{n}$ of them for some prime $p$ and some integer $n \ge 1$. Conversely, for every prime power $p^{n}$ there is a field with $p^{n}$ elements, and any two finite fields of the same size are isomorphic.

We may therefore speak of the field of order $q = p^{n}$ and write it $\mathbb{F}_{q}$, or $\mathrm{GF}(q)$.

To see why only prime powers occur, add $1$ to itself repeatedly inside a finite field. The sums $1,\ 1 + 1,\ 1 + 1 + 1, \dots$ must eventually reach $0$, and the first one that does takes a prime number $p$ of steps: a composite count would factor into two smaller nonzero sums whose product is $0$, which a field forbids. The multiples of $1$ then form a copy of $\mathbb{Z}/p\mathbb{Z}$ inside the field, the field is a vector space over this copy, and a vector space of dimension $n$ over a field with $p$ elements has exactly $p^{n}$ elements.

One further theorem completes the picture.

Theorem (Multiplicative group of a finite field).

The nonzero elements of a finite field of order $q$ form a cyclic group of order $q - 1$ under multiplication: some single element has every nonzero element among its powers.

Ways to work on it

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