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        riddles >> putnam exam (pure math) >> Fertile fields
        
(Message started by: TimMann on Nov 8^th, 2003, 12:08am)

Title: Fertile fields
Post by TimMann on Nov 8^th, 2003, 12:08am

Is there a field whose additive group is isomorphic to its multiplicative group?

I don't know the answer to this one. I found it in a sheaf of papers in my files. The person who gave it to me had written down a purported example of such a field, and this same example was the first thing that came to mind for me too, but it's wrong. I'll give this incorrect try as a sort of hint, if necessary, after folks have had time to think about the problem.

Title: Re: Fertile fields
Post by Eigenray on Nov 8^th, 2003, 5:49am

This problem was address in this thread (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_putnam;action=display;num=1044327405).
Essentially it comes down to looking at [hide]elements of order 2[/hide].

Title: Re: Fertile fields
Post by TimMann on Nov 8^th, 2003, 9:47am

Thanks -- I should have searched before posting. I think your proof in that thread could be condensed a bit, but it looks right.

The wrong example that someone had scrawled was "reals, logarithms." But that's wrong, because negative numbers don't have logarithms over the reals. It does establish an isomorphism between the additive group and a subgroup of the multiplicative group (namely the positive reals under multiplication), but that's not what was requested.

I'd gotten as far as showing that it can't work for finite fields, and that it can't work for the reals because of -1, but not as far as generalizing that to all fields.