(From Milne) If $L/F$ is an extension of fields of degree 2, then there is an automorphism $\sigma$ of $L$ such that $F$ is the fixed field of $\sigma$

Question

This is an exercise in Milne's notes.

The answer is short, it says:

(a) is false—could be inseparable. (b) is true—couldn’t be inseparable.

So may I please ask how does it related to separablity? What is the point here? May I please ask for some explicit argument with an explicit example? Thanks a lot!

Answer 1

An inseparable extension of degree $2$ is $L=F(\sqrt a)$ in characteristic $2$. As $\sqrt a$ is the unique square root of $a$ then any $F$-automorphism fixes $\sqrt a$ and so is trivial on $F$.

The classic example is the rational function field $L=\Bbb F_2(t)$ and $F=\Bbb F_2(t^2)$.

A separable extension of degree $2$ in characteristic $2$ is an Artin-Schreier extension $L=F(b)$ where $b^2+b\in F$. The nontrivial automorphism takes $b$ to $b+1$.

I am sure you know all about degree $2$ extensions in characteristics other than $2$.