Galois group of $x^{4}-4x^{2}+5$ over $\mathbb{Q}$.

Question

Determine the Galois groups of the following polynomials over the fields indicated:

(i) $x^{4}-4x^{2}+5$ over $\mathbb{Q}$

Answer 1

This can be decided by the discriminant $disc(f)$ and the cubic resolvent $R_3(f)$ of $f$, see K. Conrad's notes Galois groups of cubics and quartics, Theorem $3.6$ and following. Corollary $4.5$ shows how to find it for all irreducible quartics of the form $$ f(x)=x^4+bx^2+d. $$ The possibilities are either $V_4,D_4$ or $C_4$.