> <gayle...@eircom.net> wrote:
> >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
> >>  Z_p(x^p, y^p), but it is not a simple extension."

> >> [...]

> >> Proof:

> >> [...]

> >Seems fine to me.

> That's a relief.  Although I still couldn't see anything wrong with
> 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.
> >If u is in K then u^p is in k.
> >Hence u is of degree p (or 1).

> On that last point: the first draft of my post said that [k(u):k] =
> 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
> irreducible in K[x], and not all of the coefficients a_i are pth
> powers of elements of K."