Finite Fields

This handout discusses finite fields: how to construct them, properties of elements in a finite field, and relations between different finite fields (including their Galois groups). We write Z/(p) and Fp interchangeably for the field of size p. Here is an executive summary of the main results. • Every finite field has prime power order. • For every prime power, there is a finite field of that order. • For a prime p and positive integer n, there is an irreducible π(x) of degree n in Fp[x], and Fp[x]/(π(x)) is a field of order p n. • Any two finite fields of the same size are isomorphic (usually not in just one way). • Every finite extension of Fp is a Galois extension whose Galois group over Fp is generated by the pth power map. • If [Fp(α) : Fp] = d, the Fp-conjugates of α are α, αp, αp 2 , . . . , αp d−1 .