Let L be the splitting field of f(x)=(x^4)-2 over Q. Prove that Gal(L/Q) is a subgroup of S₄ isomorphic to D₄.
To show Gal(L/Q) is D_4, you need to verify that the resolvent cubic of f(x) factors as linear times irreducible quadratic and f(x) is irreducible over $\displaystyle \mathbb{Q}(\sqrt{D})$, where D is the discriminant of the resolvent cubic of f(x). See here or Dummit and Foote's p613-615.