Probably a straightforward question:
Let $f \in K[X]$ be a polynomial with disctinct roots $ \alpha_1, ..., \alpha_d$ in a splitting field $L$. Set $\Delta = \Pi_{i
I can see that $D$ is fixed by $\mbox{Gal}(L/K)$, but that $\Delta$ isn't necessarily. In fact, provided $\mbox{char}(K) \neq 2$, if $f$ is irreducible and separable of degree $d$, then $\mbox{Gal}(L/K)$ is a subgroup of $A_d$ iff $\Delta$ is fixed under $\mbox{Gal}(L/K)$ iff $D$ is a square in $K$.
My query is: why can't we have $\mbox{char}{K} =2$? I imagine it's something fairly obvious.
It seems this restriction is often required, which leads me to also ask: Why do results often fail when the characterstic is 2? Is it merely because these are the most trivial fields, or is there a nicer reason why we don't like them?
EDIT: Come to think of it, that results I'm thinking of that fail if $\mbox{Char}(K) = 2$ are related to quadratics in some way, so I no longer expect an answer for my second question.
Thanks