Mystic998
Feb20-08, 08:23 PM
1. The problem statement, all variables and given/known data
For prime p, nonzero a \in \bold{F}_p, prove that q(x) = x^p - x + a is irreducible over \bold{F}_p.
2. Relevant equations
3. The attempt at a solution
It's pretty clear that none of the elements of \bold{F}_p are roots of this polynomial. Anyway, so far, following a hint from the book, I've shown that if \alpha is a root of q(x), then so is \alpha + 1, and from there I was able to deduce (hopefully not incorrectly) that \bold{F}_{p}(\alpha) 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?
For prime p, nonzero a \in \bold{F}_p, prove that q(x) = x^p - x + a is irreducible over \bold{F}_p.
2. Relevant equations
3. The attempt at a solution
It's pretty clear that none of the elements of \bold{F}_p are roots of this polynomial. Anyway, so far, following a hint from the book, I've shown that if \alpha is a root of q(x), then so is \alpha + 1, and from there I was able to deduce (hopefully not incorrectly) that \bold{F}_{p}(\alpha) 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?