Re: irreducible polynomial x^{p^n} - a over a field of characteristic p

In article <1162911159.212753.237610@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"eugene" <jane1806@xxxxxxxxxx> wrote:

Here is the problem : Let k be a field of char p. Suppose a in K
doesn't have a p'th root in K. Prove that

the polynomial f(x) = x^{p^n} - a is irreducible over K.

Here is what i've done. Let b such that b^p^n = a in the splitting
field of f. Then f(x) = (x - b)^p^n.

Suppose f isn't irreducible and let g(x) be it's irreducible factor.
Then g(x) = (x - b)^k for some 1<= k <= p^n.

But then g is prima facie reducible, unless k = 1, in which case
b is in K, contradiction.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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