Let $p$ and $q$ be any two distinct integers. Write down the Galois group of $Q( √p, √q)$ over $Q$ and identify it with some known group. Determine all the sub-fields of $Q( √p, √q)$.
$Attempt$: I solve it If $p$ and $q$ are primes then it is isomorphic to $Z$/$2Z$ $*$ $Z$/$2Z$ but what about arbitrary distinct integers I don't get idea. I got that I need to consider also that what if product of $p*q$ is square or square-free.
Ok, let's start with determining the degree of the field extension to get an idea what the possible groups are. Since $$ \mathbb{Q}(\sqrt{p},\sqrt{q}) = \mathbb{Q}(\sqrt{p})(\sqrt{q}),$$ we can deduce that $$ [\mathbb{Q}(\sqrt{p},\sqrt{q}) : \mathbb{Q}]=[\mathbb{Q}(\sqrt{p},\sqrt{q}) : \mathbb{Q}(\sqrt{p})][\mathbb{Q}(\sqrt{p}) : \mathbb{Q}].$$ Both integers are zeros of the polynomial $X^2-p$ or $X^2-q$, therefore each extension can have degree of at most $2$. So the possible groups are $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ for degree $4$, $\mathbb{Z}/2\mathbb{Z}$ for degree $2$ and the trivial group for degree $1$. ($3$ is not possible since it is not a multiple of $2$).
Degree $1$ can only happen if both numbers are square (why?). So let's say $p$ is not a sqare, so $\mathbb{Q}(\sqrt{p})$ is a proper field extension of $\mathbb{Q}$ with degree $2$. (Note the degree of the minimal polynomial). If $\sqrt{q}$ is contained in either $\mathbb{Q}$ or $\mathbb{Q}(\sqrt{p})$ we are done. If not we get another field extension of degree $2$, so overall we will have a degree of $4$. Now, if this extension would have the galois group $\mathbb{Z}/4\mathbb{Z}$, we would need an element of degree $4$ in there. But note that the automorphisms only permute zeros of the same minimal polynomial, so $\sqrt{p}$ can only be mapped to itself or $-\sqrt{p}$ the same for $\sqrt{q}$. So you can write down all the automorphisms explicitlyand see, that there is no element of order $4$ and we have the group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.
You can get the subfields by looking at the different subgroups of the galois group. This shouldn't be too hard.