Degree 4 extension of $\mathbb {Q}$ with no intermediate field

Question

I am looking for a degree $4$ extension of $\mathbb {Q}$ with no intermediate field. I know such extension is not Galois (equivalently not normal). So I was trying to adjoin a root of an irreducible quartic. But I got stuck. Any hint/idea/solution?

Answer 1

With regard to Sebastian Schoennenbeck's comment, an extension of $\mathbb{Q}$ with Galois group $A_4$ (alternating group on 4 points) will do the trick. Such an extension certainly exists, in fact all alternating groups are Galois groups over $\mathbb{Q}$.