Fixed field and prime field

Question

I know that F is subset of field K. We know Any $\sigma$ in $Aut(K)$ if $\alpha$ is in F then $\alpha^{-1}$ is in F since identity goes to identity and
$\sigma(\alpha.\alpha^{-1})$= $\sigma(1) =1 $ ; $\sigma(\alpha).\sigma(\alpha^{-1})$ = $\alpha.\sigma(\alpha^{-1})$ =1 so $\sigma(\alpha^{-1})=\alpha^{-1}$ so inverse for every element exist in Fixed automorphisam. Now how to show it contains prime subfield of $K$?

Answer 1

Given any field $K$ and any subfield $F$, $F$ contains the prime subfield of $K$, since the prime subfield of $K$ is its smallest subfield.

Answer 2

Hint: The prime field of $K$ is contained in every subfield of $K$.