Computing a Galois group

Question

Let $K=\mathbb Q$ and $L$ be the splitting field of the polynomial $f(x)=x^4-6x^2+4$. I want to calculate the Galois group $\text{Gal}(L/K)$.

The zeros of $f$ are obviously

$x_{1,2}=\pm\sqrt{3+\sqrt{5}}$ and $x_{1,2}=\pm\sqrt{3-\sqrt{5}}$

Hence we have $L=\mathbb Q(x_1,x_2,x_3,x_4)$

I have the suspicion that the extension is generated by just one of them, but I can't see it.

Answer 1

Hint: $\left(\sqrt{3+\sqrt{5}}\right) \left(\sqrt{3-\sqrt{5}}\right)=?$

Answer 2

You already know the extension's Galois Group is of order four, and there aren't that many groups of that order.

You may also want to use that in fact

$$L=\Bbb Q(\sqrt2+\sqrt3)=\Bbb Q(\sqrt2\,,\,\sqrt3)\;$$

From the above, you must be able to deduce your group isn't cyclic.