I want to ask you some question on the Galois group of some polynomial.
Let $p_1<p_2<\cdots<p_n$ be distinct prime numbers.
Let's consider $f(t)=(t^2-p_1)(t^2-p_2)\cdots(t^2-p_n)\in \mathbb{Q}[t]$.
Then what is the Galois group of $f(t)$? (I am asking the shape of $Gal(E/\mathbb{Q})$ where $E$ is the splitting field of $f(t)\in \mathbb{Q}[t]$)
For $n=1,2,3$, I found out that $G_f\simeq (\mathbb{Z}_2)^n$ and so I guess it should hold for all $n$. But I can't prove it for $n \ge 4$.
How can we prove it?
