Prove that any algebraic closed field is infinite.

Definition.

A field $F$ is said to be algebraically closed if each non-constant polynomial in $F[x]$ has a root in $F$.

Proof.

Let $F$ be a finite field and consider the polynomial
\[f(x)=1+\prod_{a\in F}(x-a).\] The coefficients of $f(x)$ lie in the field $F$, and thus $f(x)\in F[x]$. Of course, $f(x)$ is a non-constant polynomial.

Note that for each $a \in F$, we have
\[f(a)=1\neq 0.\] So the polynomial $f(x)$ has no root in $F$.
Hence the finite field $F$ is not algebraic closed.

It follows that every algebraically closed field must be infinite.

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