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.

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- May 19th 2013, 01:50 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, 06:51 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, 07:14 PMProve ItRe: Show 4th degree polynomial is irreducible over Q
Or you could just do this: Let $\displaystyle \displaystyle \begin{align*} X = x^2 \end{align*}$ which makes your equation

$\displaystyle \displaystyle \begin{align*} x^4 - 22x^2 + 1 &= X^2 - 22X + 1 \\ &= X^2 - 22X + (-11)^2 - (-11)^2 + 1 \\ &= (X - 11)^2 - 120 \\ &= \left( X - 11 - 2\,\sqrt{30} \right) \left( X - 11 + 2\,\sqrt{30} \right) \\ &= \left( x^2 - 11 - 2\,\sqrt{30} \right) \left( x^2 - 11 + 2\,\sqrt{30} \right) \\ &= \left[ x^2 - \left( \sqrt{ 11 + 2\,\sqrt{30} } \right) ^2 \right] \left[ x^2 - \left( \sqrt{ 11 - 2\,\sqrt{30} } \right) ^2 \right] \\ &= \left( x - \sqrt{ 11 + 2\,\sqrt{30} } \right) \left( x + \sqrt{ 11 + 2\,\sqrt{30} } \right) \left( x - \sqrt{ 11 - 2\,\sqrt{30} } \right) \left( x + \sqrt{ 11 - 2\,\sqrt{30} } \right) \end{align*}$

Clearly none of these linear factors is rational, so your quartic can not be reduced over the rationals. - May 19th 2013, 07:29 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, 10:16 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).

$\displaystyle x^4-5x^2+6=(x-\sqrt2)(x+\sqrt2)(x-\sqrt3)(x+\sqrt3)$ with none of the linear factors in Q[x]. So the

polynomial is irreducible over the rationals? But

$\displaystyle x^4-5x^2+6=(x^2-2)(x^2-3)$ and so the polynomial is reducible in Q[x].