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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.



marked as duplicate by tomasz, Brian M. Scott, Asaf Karagila, user2468, Sasha Aug 21 '12 at 17:23

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    $\begingroup$ You may be interested in thehcmr.org/issue2_1/mfp.pdf $\endgroup$ – EuYu Aug 21 '12 at 16:41
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    $\begingroup$ The problem has come up on MSE. There is a detailed answer, with many additional references, by Bill Dubuque. Here is a link. $\endgroup$ – André Nicolas Aug 21 '12 at 16:44
  • $\begingroup$ (I hate to read "linear independance" ;-). Therefore, even though people want to close this thread, I tried to change it to "linear independence" which I was not allowed to do, since a posting with that heading already existed. For this reason I slightly changed the heading, hope this is ok for you). $\endgroup$ – user20266 Aug 21 '12 at 16:53
  • $\begingroup$ Thanks for correcting my spelling (I'm not a native english speaker)! $\endgroup$ – Sebastian Aug 21 '12 at 16:58
  • $\begingroup$ Thanks, so it seems there is not a very elementary solution. This exercise comes from a list of exercises for beginning linear algebra and the next exercise asked the same question for $\log(p)$ (that is quite easy). $\endgroup$ – Sebastian Aug 21 '12 at 17:04

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