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 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.