quasi
Posts:
10,450
Registered:
7/15/05


Re: about the KroneckerWeber theorem
Posted:
Feb 4, 2013 2:18 PM


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

