Originally Posted by

**AlexanderW** Hello.

I have to find a splitting field $\displaystyle L$ of the polynomial $\displaystyle f=x^4-4x^2+9 \in \mathbb Q[x]$ over $\displaystyle \mathbb Q$ and the galois group and all intermediate fields of the algebraic extension $\displaystyle L/\mathbb Q$.

I found out, that $\displaystyle f$ is irreducible, the roots of $\displaystyle f$ are:

$\displaystyle a=\sqrt{2-i\sqrt{5}}, \quad b=-\sqrt{2+i\sqrt{5}}, \quad c=\sqrt{2+i\sqrt{5}}=-b, \quad d=-\sqrt{2-i\sqrt{5}}=-a$ and that $\displaystyle \mathbb Q(a)$ is a splitting field of $\displaystyle f$.

Furthermore I find out, that the three homomorphisms

$\displaystyle a \mapsto \pm b, \ a \mapsto -a$ have order 2, so the galois group is isomorph to $\displaystyle \mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$.

But now I don't know how to find the fixed fields, which belongs to the homomorphisms.

For example: Which fixed field belongs to the homomorphism $\displaystyle \sigma_1: a \mapsto b=\frac{-3}{a}$ ?

The definition says:

$\displaystyle \mathbb Q(a)^{\left \{ id, \sigma_1 \right \}}=\left \{ z \in \mathbb Q(a)\ |\ \sigma_1(z)=z \right \}$

...and this don't help me, because I need a description of the fixed field like this: $\displaystyle \mathbb Q(\beta) $ (with suitable $\displaystyle \beta$).

I hope, you can help me.

Bye,

Alex