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**ZetaX** I have an irreducible polynomail f(x) = x^4 +bx^2 +d over rationals.

And I have that the Galois group of this polynomail is D_4, i.e., dihedral

group of degree 8.

Then I want to show that there are only three subfields of degree 2, and I

want to show that these are the following.

Q(d^{1/2}), Q({b^2-4d}^{1/2}), and Q({d(b^2-4d)}^{1/2}).

Since there are only three subgroups of order 4 in D_4, we have by

fundamental theorem of Galois theory that there are three subfields of

degree 2. But how can I find them, i.e., how can I show that these are the

subfields.