Algebraically Closed Field is Infinite

Theorem

Let $F$ be an algebraically closed field.

Then $F$ is infinite.

Proof

We prove the contrapositive: that a Galois field cannot be algebraically closed.

Let $F$ be Galois.

Define the polynomial:

$\ds \map f x = 1 + \prod_{a \mathop \in F} \paren {x - a}$

By definition, a field is a ring.

Thus by Ring Product with Zero:

$\ds \forall x \in F: \prod_{a \mathop \in F} \paren {x - a} = 0$

But:

$\forall a \in F: \map f a = 1 + 0 \ne 0$

Therefore $\map f x$ cannot have a root in $F$.

Thus $F$ is not algebraically closed.

The result follows by the Rule of Transposition.

$\blacksquare$