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The square roots of the primes are linearly independent over the field of rationals

I would like to prove that the family $\{\sqrt{p}, p\text{ prime number} \}$ is linearly independent in $\mathbb R$ where $\mathbb R$ is a $\mathbb Q$-vector space.

I know how to prove this for up to 4 elements but I would like a general proof as elementary as possible.

Thanks in advance.

Sebastian

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    The first inductive solution in the link I gave is about as elementary as it gets (in the sense that it doesn't require much background theory), but "elementary" and "easy" are very different things.2012-08-21

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