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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 . Then the discriminant $D$ of $f$ is $D = \Delta^2 = (-1)^{d(d-1)/2} \Pi_{i \neq j} (\alpha_i - \alpha_j) $.

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

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    Dear Jonathan, fields of characteristic 2 are a little strange but I can't answer your question "is there a nicer reason why we don't like them?" , because personally I *do* like them!2011-12-12

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