abstract algebra - Elementary proof for $sqrt{p_{n+1}} notin mathbb{Q}(sqrt{p_1}, sqrt{p_2}, ldots, sqrt{p_n})$ where $p_i$ are different prime numbers.

Take $p_1, p_2, \ldots, p_n, p_{n+1}$ be $n+1$ prime numbers in $\mathbb{P} \subseteq \mathbb{N}$. $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ seems to be quite clear, but still need a proof. I know some proofs are involved with Galois theory, which is not I want.