The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: about the Kronecker-Weber theorem
Replies: 6   Last Post: Feb 4, 2013 7:45 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 12,067
Registered: 7/15/05
Re: about the Kronecker-Weber theorem
Posted: Feb 4, 2013 2:18 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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]
>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
>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.


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.