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Math Help - Show 4th degree polynomial is irreducible over Q

  1. #1
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    Show 4th degree polynomial is irreducible over Q

    I want to show that x4-22x2+1 is irreducible over Q. I believe I need to use the Eisenstein criterion, but Im not really sure how.
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  2. #2
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    Re: 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:

    Show 4th degree polynomial is irreducible over Q-mhfrings5.png
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  3. #3
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    Re: Show 4th degree polynomial is irreducible over Q

    Quote Originally Posted by jquinc21 View Post
    I want to show that x4-22x2+1 is irreducible over Q. I believe I need to use the Eisenstein criterion, but Im not really sure how.
    Or you could just do this: Let \displaystyle \begin{align*} X = x^2 \end{align*} which makes your equation

    \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.
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    Re: 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
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    Re: Show 4th degree polynomial is irreducible over Q

    Hi Prove It,

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

    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

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