Existence of an irreducible quartic polynomial in $\mathbb{Q}[x]$ with four real roots and Galois group $A_4$.

Question

There is an example of an irreducible quartic with rational coefficients whose roots are all real and whose Galois group is $S_4$.

Is there a similar example of an irreducible quartic $f$ in $\mathbb Q [x]$ whose roots are all real and whose Galois group is $A_4$? Certainly $f$ and its resolvent cubic have to be irreducible and the discriminant should be a rational square.

Answer 1

Almost complete general results are available for cubic and quartic equations, see e.g. I. Kaplansky's booklet "Fields and rings", Part I, §10. For quartics, the main theorem reads:

Let f be a separable irreducible quartic over a field K of characteristic $\neq 2$. Let m be the degree over K of the splitting field of the resolvent cubic of f, and G be the Galois group of f over K. Then:

• if m=6, $G\cong S_4$ - if m=3, $G\cong A_4$ - if m=1, G is the Vierergruppe - if m=2, G is either of order 8 or cyclic of order 4. One way to distinguish the two cases is to determine whether f is still irreducible after the roots of the resolvent cubic are adjoined to K.