Corollary

Tag 06QA, permanent. Cite it as [Bourbaki, Tag 06QA].

— Two finite fields of the same cardinal are isomorphic.

Let $K'$ be a finite field of $q$ elements; its characteristic is a prime number $p'$ dividing $q = p^f$ , whence $p' = p$ . Therefore $K'$ is a splitting field of the polynomial $X^q - X$ in $\mathbf{F}_p[X]$ (Prop. 2), and so $K$ and $K'$ are isomorphic (V, p. 22, Cor.).

When $K = \mathbf{F}_p$ , Formula (1) reduces to the relation

X^p - X \equiv \prod_{i=0}^{p-1} (X - i) \mod p \mathbf{Z}[X]

in the polynomial ring $\mathbf{Z}[X]$ .

Formula (2) may also be written

X^{p-1} - 1 \equiv \prod_{i=1}^{p-1} (X - i) \mod p \mathbf{Z}[X]

In particular for $X = 0$ we obtain (« Wilson's formula »)

(p-1)! \equiv -1 \mod p

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 12

Cites

Prop. 2
V, p. 22