It is relatively easy to show that if $p_1$, $p_2$ and $p_3$ are distinct primes then $\sqrt{p_1}+\sqrt{p_2}$ and $\sqrt{p_1}+\sqrt{p_2}+\sqrt{p_3}$ are irrational, but the only proof I can find that $\sqrt{p_1}+\sqrt{p_2}+...+\sqrt{p_n}$ is irrational for distinct primes $p_1$, $p_2$, ... , $p_n$ requires we consider finite field extensions of $\mathbb{Q}$.
Is there an elementary proof that $\sqrt{p_1}+\sqrt{p_2}+...+\sqrt{p_n}$ is irrational exist?
(By elementary, I mean only using arithmetic and the fact that $\sqrt{m}$ is irrational if $m$ is not a square number.)
The cases $n=1$, $n=2$, $n=3$ can be found at in the MSE question sum of square root of primes 2 and I am hoping for a similar proof for larger $n$.
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