finite field



Linear algebra



A field with finitely many elements.


Let FF be a finite field. As \mathbb{Z} is the initial object in the category of rings, there is a unique ring homomorphism F\mathbb{Z} \to F, whose regular epi-mono factorization is

/(p)F;\mathbb{Z} \twoheadrightarrow \mathbb{Z}/(p) \hookrightarrow F;

here pp is prime (irreducible) since any subring of a field has no zero divisors. Here /(p)\mathbb{Z}/(p) is a prime field, usually denoted 𝔽 p\mathbb{F}_p.

Thus we have an inclusion of fields 𝔽 pF\mathbb{F}_p \to F; in particular FF is an 𝔽 p\mathbb{F}_p-module or vector space, clearly of finite dimension nn. It follows that FF has q=p nq = p^n elements.

In addition, since the multiplicative group F ×F^\times is cyclic (as shown for example at root of unity), of order q1q - 1, it follows that FF is a splitting field for the polynomial x q11𝔽 p[x]x^{q-1} - 1 \in \mathbb{F}_p[x]. As splitting fields are unique up to isomorphism, it follows that up to isomorphism there is just one field of cardinality q=p nq = p^n; it is denoted 𝔽 q\mathbb{F}_q.1

The Galois group Gal(𝔽 p n/𝔽 p)Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) is a cyclic group of order nn, generated by the automorphism σ:𝔽 p n𝔽 p n\sigma: \mathbb{F}_{p^n} \to \mathbb{F}_{p^n} sending xx px \mapsto x^p. (One way to see that σ\sigma preserves addition is to write (binomial theorem)

(x+y) p = i=0 p(pi)x iy pi = x p+y p\array{ (x + y)^p & = & \sum_{i=0}^p \binom{p}{i} x^i y^{p-i} \\ & = & x^p + y^p }

where the second equation follows from the fact that the integer pp divides the numerator of p!i!(pi)!\frac{p!}{i!(p-i)!}, but neither factor of the denominator, if 0<i<p0 \lt i \lt p. This is true for any commutative algebra over 𝔽 p\mathbb{F}_p; see freshman's dream.)

This σ\sigma is called the Frobenius (auto)morphism or Frobenius map. More generally, if mm divides nn, then 𝔽 p m\mathbb{F}_{p^m} is the fixed field of the automorphism σ m:xx p m\sigma^m: x \mapsto x^{p^m}, and Gal(𝔽 p n/𝔽 p m)Gal(\mathbb{F}_{p^n}/\mathbb{F}_{p^m}) is a cyclic group of order n/mn/m that is generated by this automorphism, which is also called the Frobenius map (for the field extension 𝔽 p n/𝔽 p m\mathbb{F}_{p^n}/\mathbb{F}_{p^m}), or just “the Frobenius” for short.

If K=𝔽 p¯K = \widebar{\mathbb{F}_p} is an algebraic closure of 𝔽 p\mathbb{F}_p, then KK is the union (a filtered colimit) of the system of such finite field extensions 𝔽 q\mathbb{F}_q and inclusions between them. If q=p nq = p^n, then 𝔽 q\mathbb{F}_q may be defined to be the fixed field of the automorphism σ n:KK\sigma^n: K \to K.

The Galois group Gal(K/𝔽 p)Gal(K/\mathbb{F}_p) is the inverse limit of the system of finite cyclic groups and projections between them; it is isomorphic to the profinite completion

^=hom(/,/) primesp p\widehat{\mathbb{Z}} = \hom(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z}) \cong \prod_{primes\; p} \mathbb{Z}_p

where p\mathbb{Z}_p is the group of pp-adic integers.


  1. There is no ‘canonical’ choice of such a splitting field, just as there is no canonical choice of algebraic closure. So ‘morally’ there is something wrong with saying ‘the’ finite field 𝔽 q\mathbb{F}_q, although this word usage can be found in the literature. In the special case n=1n = 1 there is no problem: Given fields containing pp elements are isomorphic in a unique way.

Revised on July 7, 2015 11:34:52 by Ingo Blechschmidt (