Solutions of $x^{p^n}=x$ form a subfield $F\subset K$.

Question

$K$ is an algebraically closed field such that Char $K=p$. Show that the solutions of $x^{p^n}=x$ form a subfield $F\subset K$.

My Work:

$0$ is a solution and $0\in K$. Let $\alpha$ be a non zero root. Then $\alpha^{p^n-1}=1$. I want to show that $\alpha\in K$. But stuck in showing so. Can anyone please help?

Answer 1

The fact that all roots are in $K$ follows from the fact that $K$ is algebraically closed. The fact that $F$ is a field follows from the fact that $(a+b)^{p^n}=a^{p^n}+b^{p^n}$ and similarly for multiplication. Trying to fill in the details is a good exercise I think.

Answer 2

Since char $K = p, K$ contains an isomorphic copy of $\mathbb{Z}/p\mathbb{Z}.$ Now consider $\mathbb{Z}/p\mathbb{Z}[x] \subset K[x]$ and take the splitting field of $x^{p^n} - x$ inside $K.$

Answer 3

K is algebraically closed and contains all roots of polynomials with coefficients in K.

$x^{p^n}-x=0$ has derivative $-1$ (mod $p$) so all roots are distinct. if $\alpha$ and $\beta$ are roots then the fact that $$ (\alpha+\beta)^{p^n} \equiv_p \alpha^{p^n}+\beta^{p^n} \\ (\alpha\beta)^{p^n} \equiv_p \alpha^{p^n}\beta^{p^n} $$ means the roots form a ring.

if $\alpha \ne 0$ then $\alpha^{p^n-1} \equiv 1$ so $\alpha$ has an inverse