Date: Feb 4, 2013 2:18 PM
Author: quasi
Subject: Re: about the Kronecker-Weber theorem
David Bernier wrote:

>

>Let's suppose the base field is Q, and P(x) is an irreducible

>polynomial of degree n over Q. Let alpha_1, ... alpha_n

>be the n conjugate roots in the splitting field L (subfield of

>C, the complex numbers) of P(x) over Q.

>

>If sigma: {alpha_1, ... alpha_n} -> {alpha_1, .. alpha_n}

>is a permutation of the n conjugate roots,

>

>then according to me if a field automorphism of phi of L exists

>which acts on {alpha_1, ... alpha_n} the same way the

>permutation sigma does,all the elementary symmetric polynomials

>in n indeterminates must be invariant under the application of

>such elementary symmetric polynomials:

>

>[wikipedia, with def. of elementary symmetric polynomials]

>

>http://en.wikipedia.org/wiki/Elementary_symmetric_polynomial

>

>In the other direction, if we have a sigma, permutation as above,

>and all the elementary symmetric polynomials are left

>invariant, does it follow that for the splitting field L,

>there is a field automorphism phi of L such that

> phi(alpha_j) = sigma(alpha_j), 1<=j<=n ?

>In other words, phi acts on the alpha_j the same way sigma

>does.

>

>If the elementary symmetric polynomials are left invariant

>by sigma, does it follow that some automorphism phi of L

>acts on {alpha_1, ... alpha_n} the same way sigma acts ?

The elementary symmetric functions of the roots are

left invariant by _any_ permutation of the roots.

quasi