Let $ p_1,p_2,\ldots,p_n $ be distinct prime numbers. Prove that the polynomial $$ f(x)=\prod_{e_1,e_2,\ldots,e_n=\pm1}(x+e_1\sqrt{p_1}+e_2\sqrt{p_2}+\cdots+e_n\sqrt{p_n}) $$ Is irreducible in $ \mathbb Z[x] $.
This means that the polynomial $ f $ is the minimal polynomial of the number $ \sqrt{p_1}+\sqrt{p_2}+\cdots+\sqrt{p_n} $ over the field of rational numbers!