Solution
= Solution
Factor $f_1=(X^2-3)(X^2+2)$. Its <splitting field> is $L=\mathbb Q(\sqrt3,\sqrt{-2})$. The first <quadratic field> is real, so it cannot contain the nonreal $\sqrt{-2}$. Thus adjoining the second square root doubles the degree again. Independent sign changes of the two square roots give all four <automorphisms>, so
$$
\boxed{[L:\mathbb Q]=4,\qquad\operatorname{Gal}(L/\mathbb Q)\cong C_2\times C_2.}
$$