abstract algebra - when can I get a Galois element from the permutation of the roots?

Given prime numbers $p_1, \cdots, p_n$, Define $E:= \mathbb{Q} [\sqrt{p_1}, \cdots, \sqrt{p_n}]$ a Galois extension over $\mathbb{Q}$ with the separable polynomial

$$p(x) = \prod(x^2-p_i).$$

I know in general, permuting roots does not always give us a Galois group element.

In this case, if I have the permutation $\sqrt p_1 \mapsto -\sqrt p_1$ and fixes other $\sqrt p_i$'s, we want to define an element in $Gal(E/\mathbb{Q})$ from this, what do we need to check?

I guess we have to check that $\sqrt p_1 \not\in \mathbb{Q}[\sqrt p_2, \cdots, \sqrt{p_n}]$, and is there anything else that I need to check?

Edit: This is the exercise 18.13 from M. Isaacs. The first part is to show $Gal(E/\mathbb{Q}) = \left(Z_2\right)^n$. This is the second part, and the next part is to show $\sqrt p_1, \cdots, \sqrt p_n$ are linearly independent. So maybe showing $\sqrt p_1 \not \in \mathbb{Q}[\sqrt p_2,\cdots \sqrt p_n]$ is not easy.