It is relatively easy to show that if p1, p2 and p3 are distinct primes then √p1+√p2 and √p1+√p2+√p3 are irrational, but the only proof I can find that √p1+√p2+...+√pn is irrational for distinct primes p1, p2, ... , pn requires we consider finite field extensions of Q.
Is there an elementary proof that √p1+√p2+...+√pn is irrational exist?
(By elementary, I mean only using arithmetic and the fact that √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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