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

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David Bernier

Posts: 3,222
Registered: 12/13/04
Re: about the Kronecker-Weber theorem
Posted: Feb 4, 2013 2:05 PM
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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 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.



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