(a) Find a splitting field $\displaystyle K$ for the polynomial $\displaystyle f(x)=(x^2-3)(x^2-5) \in \mathbb{Q}[x]$.

(b) Determine the Galois Group of $\displaystyle K$ over $\displaystyle \mathbb{Q}$.

(c) Find all intermediate fields between $\displaystyle \mathbb{Q}$ and $\displaystyle K$, by refering to the Fundamental Theorem of Galois Theory [FTG] to show you have them all.

I'm pretty sure that (a) is $\displaystyle \mathbb{Q}(\sqrt(3), \sqrt(5))$, (b) is $\displaystyle \mathbb{Z}_2 \times \mathbb{Z}_2$ (c) $\displaystyle \mathbb{Q}, \mathbb{Q}(\sqrt(3), \sqrt(5)), \mathbb{Q}(\sqrt(3)), \mathbb{Q}(\sqrt(5)), \mathbb{Q}(\sqrt(15))$. I don't know how to justify (c) by using the Fundamental Theorem of Galois Theory. Also, I don't know how to determine the Galois Group for part (b). I know it is either $\displaystyle \mathbb{Z}_2 \times \mathbb{Z}_2$ or $\displaystyle \mathbb{Z}_4$. But, I don't know how to show this.