Finding a polynomial with given Galois group In Galois theory we have just been introduced to the fundemental theorem. We were then assigned this homework problem as a sort of inverse problem to the typical "Find the Galois group of $f$ ":

Does there exist a degree 6 irreducible polynomial in $\mathbb Q[x]$ whose Galois group is isomorphic as an abstract group to $S_3$, the symmetric group of degree 3?

I already know that $x^3-2$ has Galois group $S_3$ and therefore it would suffice to find another polynomial whose splitting field is $(\sqrt[3]2, \zeta)$ with $\zeta$ the third root of unity. But at this point I get stuck. My intuition is that "nearly any" polynomial whose roots lie in this field will work, but I can't figure out how to choose the roots such that the original polynomial sits back in $\mathbb Q[x]$, except in trivial ways that will make it reducible.
Can someone give me a nudge in the right direction?

This is a primitive element issue : for your polynomial, take the minimal polynomial of $\sqrt[3]{2}+\zeta$ (in fact $\sqrt[3]{2}+t\zeta$ works for any $t\in{\mathbb Q}$, except for a few exceptional values of $t$)
