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Topic: Independent sqaure roots???
Replies: 17   Last Post: May 1, 2003 9:50 AM

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Paul Pollack

Posts: 65
Registered: 12/13/04
Re: Independent sqaure roots???
Posted: Apr 30, 2003 6:14 PM
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"Chris Hughes" <hughes@post.tau.ac.il> wrote in message
news://4e5b4dbf.0304301333.7f51f5c2@posting.google.com...
> David C. Ullrich <ullrich@math.okstate.edu> wrote in message
news:<91cvav4jl262v76q6elt4fh5oj3hrknub5@4ax.com>...
> > Are the square roots of the positive square-free integers
> > (including 1) linearly independent over Q?
> >

>
> <snip>
>
> I see that this has already been answered above with a proof, but I
> have previously seen this result quoted as "a classical result of
> Besicovich, [which states] that the square roots of distinct
> square-free integers are linearly independent over the rationals".
> This comes from page 405 of a paper of Heath-Brown in Acta
> Arithmetica, 60 (1992) 389--415. Unfortunately, he doesn't give a
> reference to the original Besicovich paper.
>
> Chris Hughes


I haven't looked it up to make sure, but I'd guess it is the following:

A.S. Besicovitch (1940). On the linear independence of fractional powers of
integers. J. London. Math. Soc. 15, 3-6.

HTH,
Paul




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