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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- Feb 18th 2009, 09:13 AMpeteryellowgalois 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.

- Feb 18th 2009, 09:23 AMpeteryellow
it should be same splitting field

soory I forgor to write - Feb 18th 2009, 06:28 PMThePerfectHacker
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 happens to have Galous group equal to (this can be shown using what your favorite method is). Thus, we require splitting field for to be a degree extension. Now, the equation solves into where . Therefore, the splitting field is . We require that . However, . But because solves . Likewise, . Therefore, and so it seems it is impossible.

- Feb 19th 2009, 10:20 AMThePerfectHacker
I think it is a good exercise to actually find the conditions on that determine what Galois group actually is.

This should not be too bad since it appears as an exercise problem in my book. - Feb 22nd 2009, 05:23 AMpeteryellow
which book are you using?

- Feb 22nd 2009, 07:48 AMThePerfectHacker
This one.