nLab finite field

Idea

A field with finitely many elements.

Structure

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

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

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

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

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

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

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

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

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

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

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

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

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

Related concepts

• modular arithmetic, number theory, field of one element

References

• Wikipedia, Finite field
• James Ax, The elementary theory of finite fields, Annals of Math. Series 2. 88 (2): 239–271 (1968) doi

category: algebra

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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 $\mathbb{F}_q$, although this word usage can be found in the literature. In the special case $n = 1$ there is no problem: Given fields containing $p$ elements are isomorphic in a unique way. ↩