Fundamental Algebra Theorem un other fields

Question

Studying Galois Theory, I came up with a question that I don't know if it is true: does the fundamental algebra theorem work on other fields different from \mathbb{R} or \mathbb{C}? In particular in finite fields?

Answer 1

The fundamental theorem of algebra is valid in every real closed field with the same statement:

If $F$ is a real closed field, then $F(\sqrt{-1})$ is algebraically closed.

The proof is based on two facts familiar for real numbers:

• Every polynomial with odd degree has a root.
• Every non-negative number has a square root.

One of the characterizations of real closed fields is:

$F$ is a real closed field iff $F$ is not algebraically closed but $F(\sqrt{-1})$ is algebraically closed.

Also:

$F$ is a real closed field iff $F$ is not algebraically closed but its algebraic closure is a finite extension.

In this sense, the result for real closed fields is the best possible analog of the fundamental theorem of algebra.

Answer 2

Every field has an extension that is algebraic closed, in the sense that any polynomial written with coefficients in the same field will have roots in that field. But no finite field is algebraically closed, as pointed out above.

The algebraic closure of a finite field of characteristic $p$ is $\bigcup_{n=1}^{\infty} \mathbb{F}_{p^{n!}}$. That is, $\mathbb{F}_{p^1}\hookrightarrow \mathbb{F}_{p^{2}}\hookrightarrow \mathbb{F}_{p^{3!}}\hookrightarrow \mathbb{F}_{p^{4!}}\hookrightarrow\cdots$. At each step it is a finite field, it is only in the limit that you get infinite, so it is "just infinite".