```Date: Feb 4, 2013 2:05 PM
Author: David Bernier
Subject: Re: about the Kronecker-Weber theorem

On 02/04/2013 04:04 AM, quasi wrote:> David Bernier wrote:>>>> I have a further question about conjugate roots ...>>>> The non-trivial third roots of unity>> -1/2 +i*srqrt(3)/2 and -1/2 -i*srqrt(3)/2>> are complex conjugates.>>>> I don't know of a definition where, for example, in the>> setting above,>> 2^(1/3) is said to be conjugate to>> 2^(1/3) * (-1/2 +i*srqrt(3)/2).>> Let H be an algebraically closed field, and let K be a> subfield of H. Let K[x] denote the ring of all polynomials> in the indeterminate x with coefficients in K. If f in K[x]> is irreducible in K[x], the roots of f in H are said to be> conjugates of each other over K.>> Thus, the 3 cube roots of 2 are conjugate over Q since> they are roots of the polynomial x^3 - 2 which is irreducible> over Q.Thank you, quasi.Let's suppose the base field is Q, and P(x) is an irreduciblepolynomial of degree n over Q.  Let alpha_1, ... alpha_nbe the n conjugate roots in the splitting field L (subfield ofC, 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 existswhich acts on {alpha_1, ... alpha_n} the same waythe permutation sigma does,all the elementary symmetric polynomials in n indeterminatesmust be invariant under the application of suchelementary symmetric polynomials:[wikipedia, with def. of elementary symmetric polynomials]http://en.wikipedia.org/wiki/Elementary_symmetric_polynomialIn the other direction, if we have a sigma, permutation as above,and all the elementary symmetric polynomials are leftinvariant, 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 bysigma, does it follow that some automorphism phi of Lacts on {alpha_1, ... alpha_n} the same way sigma acts ?David Bernier-- dracut:/# lvm vgcfgrestoreFile descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID 993: sh   Please specify a *single* volume group to restore.
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