I was playing around with finite fields, and noticed that if $F$ is a finite field, and $E$ an extension of odd degree, then every nonsquare $a\in F$ is again a nonsquare in $E$. I found this out by looking at the order of the coset of a possible root of $a$ in the quotient group $E^\times/F^\times$.
I now wonder if $E$ is an extension of even degree, does this lead every element of $F$ to have a square root in $E$? (If possible, is there a way to do so without heavy use of Galois theory?) Thanks.