Degree 2 Field extensions 
Are all degree $2$ field extensions Galois?   

I know that this is true over the rationals. But is it true in general?

No, It's not true in general. 
Here is a counterexample. Let $F$ a field of characteristic 2, and consider the field $k$ of rational functions over $F$, i.e. $k=F(T)$. Now consider the extension of degree 2 obtained adding the roots of the polynomial $x^2-T=0$. This extension is not separable (remember that $F$ so $k$ has characheristic 2) hence not a Galois extension (this is a result of Artin).
