Re: Finite extension of a finite field

On Fri, 15 Aug 2008 08:57:48 -0700 (PDT), sanchopancho80@xxxxxx wrote:

On 15 Aug., 17:49, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:
On Fri, 15 Aug 2008 08:44:00 -0700 (PDT), sanchopanch...@xxxxxx
wrote:

I've seen somebody writing F_p[a^(1/p)] with a in the finite field
F_p. What does this mean? More precisely: What does a^(1/p) mean?

Just guessing: it means F_p[b], where b is a root of x^p - a in an
extension field.

I have thought that too at first, but why are the results isomorphic
for different roots? The polynomial x^p - a doesn't have to be
irreducible.

Dashing off to the shops before they close, but I think either the
polynomial splits in F_p, or else it is irreducible ... something's
screwy here, because x^p = x for all x in Z_p, so I'm not sure what
F_p means. No time to think about it! I think the irreducibility
of x^p - a over a field K of characteristic p depends only on the
fact that a is not a pth power in K, but I'll have to check later
(by which time someone else will have clarified - but I need the
practice!).

--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.