Proposition 3

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

Algebra, V, p. 20, § 4. Algebraically closed extensions

Every algebraically closed field is infinite.

Let K be a finite field and put $f(X) = 1 + \prod_{a \in K} (X - a)$ . The polynomial $f \in K[X]$ is non-constant and $f(a) = 1$ for each $a \in K$ . So the field K does not satisfy (AC') and hence is not algebraically closed.

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

Cited by

Theorem 1, Appendix 3

Cites

Nothing.