field theory - $sqrt{p_1}$ is not in $Q[sqrt{p_2},...,sqrt{p_n}]$

How to show $\sqrt{p_1}$ is not in $Q[\sqrt{p_2},...,\sqrt{p_n}]$ if $p_1,...,p_n$ are distinct primes? Intuitively, this is pretty clear, but it makes me very uncomfortable to just believe. Any idea to prove this rigorously? I want this result because I am trying to compute the Galois group of $(X^2-p_1)...(X^2-p_n)$. If I know the statement is true, then the Galois group of this polynomial will be direct product of separate Galois group.

Answer

You may also go through the following lines: by quadratic reciprocity and Dirichlet's theorem, there is some uber-huge prime $P$ for which $p_2,p_3,\ldots,p_n$ are quadratic residues, while $p_1$ is not. It follows that the algebraic numbers $\sqrt{p_1}$ and $\sqrt{p_2}+\ldots+\sqrt{p_n}$ have different degrees over $\mathbb{F}_P$ ($2$ and $1$, respectively), so they cannot be linearly dependent over $\mathbb{Q}$.