I want to show that x^{4}-22x^{2}+1 is irreducible over Q. I believe I need to use the Eisenstein criterion, but Im not really sure how.

Printable View

- May 19th 2013, 02:50 PMjquinc21Show 4th degree polynomial is irreducible over Q
I want to show that x

^{4}-22x^{2}+1 is irreducible over Q. I believe I need to use the Eisenstein criterion, but Im not really sure how. - May 19th 2013, 07:51 PMjohngRe: Show 4th degree polynomial is irreducible over Q
Hi,

You're right. Use the Eisenstein criterion to show the polynomial has no linear factors. Then proceed:

Attachment 28414 - May 19th 2013, 08:14 PMProve ItRe: Show 4th degree polynomial is irreducible over Q
- May 19th 2013, 08:29 PMjquinc21Re: Show 4th degree polynomial is irreducible over Q
Ah, I knew I was missing something. Usually our Eisenstien problems are simpler, so I figured there would be a nice trick.

By the way, I love your name :D - May 20th 2013, 11:16 AMjohngRe: Show 4th degree polynomial is irreducible over Q
Hi Prove It,

I don't quite follow your argument (at least the last few lines).

with none of the linear factors in Q[x]. So the

polynomial is irreducible over the rationals? But

and so the polynomial is reducible in Q[x].