Let be a field of characteristic . Prove that either is irreducible or splits in ...

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- Aug 25th 2009, 11:42 PMynjSplitting Field
Let be a field of characteristic . Prove that either is irreducible or splits in ...

- Aug 26th 2009, 10:00 AMnirax
if you prove one thing you are through ...

all roots of this polynomial are equal (in some appropriate extension) - Aug 26th 2009, 10:14 AMNonCommAlg
i'll prove something more than what the problem is asking! i'll show that either is irreducible or for some :

suppose is not irreducible. then for some polynomials where is irreducible with and

differentiating (formally) will give us: hence which is impossible unless and

because so suppose and then and hence letting

we'll get