
Re: about the KroneckerWeber theorem
Posted:
Feb 4, 2013 2:05 PM


On 02/04/2013 04:04 AM, quasi wrote: > David Bernier wrote: >> >> I have a further question about conjugate roots ... >> >> The nontrivial 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 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 ?
David Bernier
 dracut:/# lvm vgcfgrestore File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID 993: sh Please specify a *single* volume group to restore.

