From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: looking for a polynomial example
Date: Mon, 05 Jun 2000 10:16:58 GMT
Newsgroups: sci.math.research
Summary: [missing]

In article <Pine.GSO.4.21.0006021824130.10590-100000@pollux.usc.edu>,
  qcheng <qcheng@pollux.usc.edu> wrote:
> I am looking for an irreducible polynomial f(x) of degree n
> having following property.
> 1. It is abelian.
> 2. Order of its galois group is greater than n.
>
> Any help will be appreciated,
> -Qi
>
> [ Moderator's note: by (1.), qcheng means that the Galois group
> is abelian.

I don't think there is one. Let G be the Galois group and
a_1, a_2, ..., a_n be the zeros of f. Then G acts transitively and
faithfully on A = {a_1, ..., a_n}. Let H be the stabilizer of a_1.
I claim that H is trivial. The stabilizer of a_j is a conjugate of
a_1 (by an element of G taking a_1 to a_j). As G is abelian then
H is the stabilizer of a_j. As this is true for each j, H is trivial
and so |G| = n.

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
 "`The twenty-first century didn't begin until a minute
  past midnight January first 2001.'"
   John Brunner, _Stand on Zanzibar_ (1968)


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