From: Robin Chapman Subject: Re: looking for a polynomial example Date: Mon, 05 Jun 2000 10:16:58 GMT Newsgroups: sci.math.research Summary: [missing] In article , qcheng 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) Sent via Deja.com http://www.deja.com/ Before you buy.