Re: Galois theory question / roots of unity
achava_at_hotmail.com
Date: 03/29/05
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Date: 29 Mar 2005 12:33:38 -0800
dave161716@hotmail.com wrote:
> (From the Stanford Algebra Qualifying Exam,
> Fall 2003)
>
> Let g(X) in Z[X] be a monic polynomial with
> roots a_1, a_2, ..., a_n, in C.
>
> Assume that |a_j| = 1, for j=1,...,n, i.e. all
> roots have absolute value equal to 1.
>
> a) Prove that each a_j is a root of unity.
>
> b) Give an example to show that a) is false if
> the assumption that g(X) be monic is dropped.
>
> I didn't get very far. Here's what little I got.
>
> Note that it is no loss of generality to assume
> that all a_j are nonreal. (For, the only real
> numbers with absolute value = 1 are 1 and -1,
> and if these occur, just divide out by X - 1
> respectively X + 1.)
>
> Now |a_j| = 1 means a_j = exp(2*\pi*i*t_j),
> where t_j is a real number, 0 <= t_j < 1.
> a_j is then a root of unity if and only if
> t is rational.
>
> Also note that if a is a root of g(X), then
> the complex conjugate is also a root of g(X).
> So the t_j's come in pairs and so we may assume
> that n=2m is even and that
> t_{m+1} = -t_1
> ...
> t_{n} = -t_m
>
> For a) I tried a proof by induction on n, the
> number of roots of g(X).
>
> For n = 1, the claim is obvious.
>
> For n > 1, assume that all t_j are irrational.
> Aim for contradiction. Now what?
>
> I suspect one may not be able to do this in
> elementary terms, i.e. w/o using some "big"
> theorem from Fields or Galois theory. Only
> I don't know which theorem might do it.
>
> Any hints would be appreciated.
>
> Thanks in advance.
>
> --Dave
Given the restriction on the complex norm of the a_i, which I prefer to
think of as the conjugates of a single root called alpha, can you think
of a way to bound the coefficients of
g(x)?
Can you think of any other numbers, that is other than alpha and its
conjugates, perhaps related in some way to alpha, whose conjugates are
also all of complex norm 1?
Does the spirit of Dirichlet and his flock of dirty pigeons hover
uncomfortably in the deep recesses of your thinking?
When the clock strikes midnight and the wolves howl and the spirits
prowl, perhaps the answer will suddenly be revealed to you.
Regards,
Achava
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