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Irreducible polynomial and Isomorphism

Hi there, I'm really stuck on this question, both parts i) and ii). As ii) rely's on i)

Im able to show that x^2-2 is irreducible over Z[5] but am stuck in trying to show that it has a zero in F. I know that I need to use (r+sa)^2-2=0 and finding r and s but I can't factorize the polynomial adequately in Z[5]... any help would be great! And let me know if im on the wrong track?

Thanks

Re: Irreducible polynomial and Isomorphism

Quote:

Originally Posted by

**bazzel** Hi there, I'm really stuck on this question, both parts i) and ii). As ii) rely's on i)

Im able to show that x^2-2 is irreducible over Z[5] but am stuck in trying to show that it has a zero in F. I know that I need to use (r+sa)^2-2=0 and finding r and s but I can't factorize the polynomial adequately in Z[5]... any help would be great! And let me know if im on the wrong track?

Thanks

I think you need to check whether there exists

$\displaystyle \alpha=r+sa\in \mathbb{Z}_5(a)\,\,s.t.\,\,\alpha^2-2=0\iff (r+sa)^2-2=0\iff r^2+2rsa+s^2a^2-2=0\iff r^2+2rsa+3s^2-2=0$.

Now choose $\displaystyle r=1\Longrightarrow 1+2sa+3s^2-2=0\iff 3s^2+2as-1=0\Longrightarrow s_{1,2}=-2a\pm \sqrt{-1}=-2a\pm 2$

For example one root is $\displaystyle r+sa=1+(2-2a)a=1+2a-2a^2=2a$

Nemesis