Does every algebraically closed field with nonzero characteristic have a unique finite subfield $p^n$?

Question

Let $F$ be an algebraically closed field such that $char(F)\neq 0$.

Then, $\forall n\in\mathbb{Z}^+$, there exists a unique finite subfield $K$of $F$ such that $|K|=char(F)^n$.

Is this statement true?

My text only introduces this for the algebraic closure of $\mathbb{Z}_p$, but it seems like the same proof works for any algebraically closed field.

Is this true?

Answer 1

Let $F$ be the field you are talking about, let $p$ be the characteristic of $F$. Then since $F$ is algebraically closed, the equation $$ x^{p^{n}}-x=0 $$ has precisely $p^{n}$ solutions in $F$ because its derivative is $1$. The solutions would now constitute a subfield $F_{n}$ of $F$. Multiplication and inverse is trivial, and addition follows from the characteristic $p$ property.

Answer 2

Yes. Any algebraically closed field of characteristic $p$ contains $\mathbb Z/p\mathbb Z$, so it must contain the algebraically closure of $\mathbb Z/p\mathbb Z$, and the subfield of size $p^n$ must be all the roots of $x^{p^n}-x$.

So once you know the result for the algebraic closure of $\mathbb Z_p$, you know it for all algebraically closed fields of characteristic $p$.