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Math Help - galois Theory

  1. #1
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    galois Theory

    Determine, if there are rational numbers c,d so that the polynomails x^4 -4x-1 and x^4+ax^2+b have splitting field over Q(rational numbers), If yes find a and b.
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  2. #2
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    it should be same splitting field

    soory I forgor to write
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  3. #3
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    Quote Originally Posted by peteryellow View Post
    Determine, if there are rational numbers c,d so that the polynomails x^4 -4x-1 and x^4+ax^2+b have splitting field over Q(rational numbers), If yes find a and b.
    I think it is impossible. This is just an idea that I have that might work. We make a trivial observation, if they have the same splitting field then the polynomials definitely have the same Galois group. The polynomial x^4 - 4x - 1 happens to have Galous group equal to S_4 (this can be shown using what your favorite method is). Thus, we require splitting field for x^4 + ax^2 + b to be a degree 24 extension. Now, the equation x^4 + ax^2 + b=0 solves into x = \pm \sqrt{ \frac{-a+r}{2}}, \pm \sqrt{ \frac{-a-r}{2}} where r = \sqrt{a^2 - 4b}. Therefore, the splitting field is K=\mathbb{Q} \left( \sqrt{ \frac{-a+r}{2}}, \sqrt{\frac{-a+r}{2}} \right). We require that [K:\mathbb{Q}]=24. However, [K:\mathbb{Q}] \leq \left[\mathbb{Q}\left( \sqrt{\frac{-a+r}{2}} \right):\mathbb{Q} \right]\left[ \mathbb{Q}\left( \sqrt{\frac{-a-r}{2}} \right) : \mathbb{Q} \right]. But \left[\mathbb{Q}\left( \sqrt{\frac{-a+r}{2}} \right):\mathbb{Q} \right] \leq 4 because \sqrt{\frac{-a+r}{2}} solves r^2 = (2x^2 + a)^2. Likewise, \left[\mathbb{Q}\left( \sqrt{\frac{-a+r}{2}} \right):\mathbb{Q} \right] \leq 4. Therefore, [K:\mathbb{Q}] \leq 16 < 24 and so it seems it is impossible.
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  4. #4
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    I think it is a good exercise to actually find the conditions on a,b that determine what Galois group x^4 + ax^2 + b actually is.
    This should not be too bad since it appears as an exercise problem in my book.
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    which book are you using?
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  6. #6
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    Quote Originally Posted by peteryellow View Post
    which book are you using?
    This one.
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