# Homework Help: Irreducible polynomial on polynomial ring

1. Nov 8, 2005

### SN1987a

How would I prove that $x^2+1$ is irreducible in $Z_p[x]$, where p is an odd prime of the form 3+4m.

I know that for it to be rreducible, it has to have roots in the ring. So $x^2=-1 (mod p)$. Or $x^2+1=k(3+4m)$, for some k. I tried induction on m, but it does not work because [itex}x^2+1[/itex] is only reducible on $Z_p[x]$ if p is prime, which is not the case for all m. Apperently, there exists a two-line solution.

Any tips would be appreciated.

2. Nov 8, 2005

### Icebreaker

Hint: Fermat's little theorem and this lemma: if R is a commutative ring with identity, and a in R is invertible, then a^n=1 and a^m=1 => a^gcd(n,m)=1.

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