I have a problem and the problem is:
Let $F$ be a field and char $F$=2.Let $$E=F[x]/(x^2+bx+c)~~b,c\in F.$$
When $b$ and $c$ satisfy what condition, $E$ is a Galois extension with $[E:F]=2$.
My opinion:
Firstly $x^2+bx+c$ should an irreducible polynomial in $F[x]$.
Then in $E$ , $x^2+bx+c$ should have two different roots.
$E$ is splitting field over $F$ with a separable polynomial, so $E/F$ is a Galois extension with $[E:F]=2$
But how to find the specific conditions which $b$ and $c$ should satisfy?
Thanks!