Let $K\supset F$ be a finite field extension of degree 2 and the characteristic of $F$ is not 2. Show that there exists an isomorphism of rings $\sigma:K\to K$ such that
$$F=\{x\in K:\sigma(x)=x\}.$$
In this case, I didn't see why we need the condition char $F\neq 2$. The extension has degree 2 this means this extension is algebraic. Then I am thinking of using the polynomial ring over $F$. Then I get stuck. Any hints?
Hint: Is there a $w\in K\setminus F$ such that $w^2 \in F$? (here you will need the hypothesis ${\text {char}}(F)\neq 2$ and $[K:F]=2$). Now, the last hint is: think about the automorphisms of ${\mathbb Q}(\sqrt{2})$, relate it to your problem.