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Irreducibility of a general polynomial in a finite field

by Mystic998
Tags: field, finite, irreducibility, polynomial
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Feb20-08, 07:23 PM
P: 206
1. The problem statement, all variables and given/known data

For prime p, nonzero [itex]a \in \bold{F}_p[/itex], prove that [itex]q(x) = x^p - x + a[/itex] is irreducible over [itex]\bold{F}_p[/itex].

2. Relevant equations

3. The attempt at a solution

It's pretty clear that none of the elements of [itex]\bold{F}_p[/itex] are roots of this polynomial. Anyway, so far, following a hint from the book, I've shown that if [itex] \alpha[/itex] is a root of q(x), then so is [itex]\alpha + 1[/itex], and from there I was able to deduce (hopefully not incorrectly) that [itex]\bold{F}_{p}(\alpha)[/itex] is the splitting field for q(x) over the field, but I can't figure out the degree of the extension, so I'm kind of stuck. Any ideas?
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Feb20-08, 07:28 PM
Sci Advisor
HW Helper
P: 2,020
How about showing that if q(x) is reducible, then it must split into linear factors?
Feb20-08, 08:56 PM
P: 206
I sat and thought about it for about 45 minutes and played around on a chalkboard with it, and I'm not sure how to go about your suggestion. Is it related to the work I already did, or is it a completely new line of attack? Thanks for helping.

Feb20-08, 10:28 PM
P: 206
Irreducibility of a general polynomial in a finite field

Okay, I figured it out. Thanks a lot.

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