Here's a way to screw yourself up

[sniffnoy]

Work on a problem that requires you to think about both matrices and finite field extensions, and as a result accidentally end up thinking the minimal polynomials of the former must be irreducible. Get horribly contradictory results.

Comments

[sniffnoy.livejournal.com]

No, I said the former. Minimal polynomials of matrices. And not of matrices given by multiplication by an element of the extension or something. :P

Also separability is irrelevant and finite fields are perfect anyway? :-/

[sniffnoy.livejournal.com]

But... what? Yes you can! Assuming by "separable" you mean "perfect", well, they are. They're finite, so since the Frobenius endomorphism is injective, it must be surjective. And finite extensions are generated by one element - namely any generator for the unit group. I guess that follows from perfectness as well, but that's not something I know. (My field theory is pretty rusty and I've been reviewing it recently.)