Example of a field extension of degree 2 which is not Galois

Question

What is an example of a field extension of degree $2$ which is not Galois? By definition, if $E/F$ is Galois, then $E$ is finite, separable, and normal over $F$. We know that every field extension of degree $2$ is normal, so we have to find a field extension that is inseparable.

Answer 1

The most typical example of a non-separable extension is $\Bbb F_p(t^{1/p})/\Bbb F_p(t)$ for a prime $p$; this has degree $p$ so if you take $p=2$ you have your answer.

Answer 2

Your example will need to be in characteristic $2$, otherwise quadratic extensions are always Galois.

If $K$ has characteristic $2$, then there are two kinds of quadratic extensions:

• $L=K(\sqrt{a})$, where $a\in K$ is not a square, is always inseparable;

• $L=K(\phi^{-1}(a))$ with $\phi(x)=x^2+x$, where $a\in K$ is not on the image of $\phi$, is always separable.

Note that of course the first case can never happen if all elements of $K$ are squares, ie if $K$ is perfect.

You can take any example of the first kind, such as the one given by Alex Mathers, which is the standard one.