How to determine if a number is quadratic residue over binary field?

Question

I am trying to determine if $x$ is quadratic residue over a binary field ($GF(2^n)$). For finite field of type $GF(p)$ (where $p$ is prime), one can find the answer by calculating $x^{(p-1)/2}$ and checking whether it is congruent to $1$ mod $p$, but I don't know how to relate this method to the binary field. Is there any better way apart from this?