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Newsgroups: sci.math
From: Angus Rodgers <twir...@bigfoot.com>
Date: Thu, 24 Jul 2008 13:47:20 +0100
Local: Thurs, Jul 24 2008 8:47 pm
Subject: Re: Finite, non-simple field extension: is this proof OK?
On Thu, 24 Jul 2008 11:45:45 +0100, Timothy Murphy
<gayle...@eircom.net> wrote: That's a relief. Although I still couldn't see anything wrong with >Angus Rodgers wrote: >> Exercise 93 in Joseph Rotman's /Galois Theory/ (2nd ed. 1998): >> "Show that Z_p(x, y) is a finite extension of its subfield >> [...] >> Proof: >> [...] >Seems fine to me. the proof, I was getting more and more worried, because it doesn't actually use any Galois theory! >The extension [K:k] is of degree p^2. On that last point: the first draft of my post said that [k(u):k] = >If u is in K then u^p is in k. >Hence u is of degree p (or 1). p or 1, but then I began to worry that it might not be true, or at least wasn't as "obvious" as I'd imagined, and I didn't need it, so I left it out, giving [k(u):k] <= p < p^2 instead. Now that I'm obliged to think about it, I think I've adapted a solution to an earlier exercise (no. 75) to prove it. I was going to post this, too - because I was worried that I was making unnecessarily heavy weather of something that might perhaps, after all, be "obvious" - but I see there's a standard result that has it as a special case: From D. J. H. Garling, /Galois Theory/ (1986), p. 88: "Theorem 10.8 Suppose that char K = p > 0 and that f(x) = g(x^p) = a_0 + a_1x^p + ... + x^{np} is monic; then f is irreducible in K[x] if and only if g is -- You must Sign in before you can post messages.
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