```Date: Feb 4, 2013 3:01 AM
Author: David Bernier
Subject: Re: about the Kronecker-Weber theorem

On 02/03/2013 08:03 PM, quasi wrote:> David Bernier<david250@videotron.ca>  wrote:>>>> The Kronecker-Weber theorem characterizes abelian extensions>> of Q.>>>> If we look at p(X) = X^3 - 2 over Q, then according to>> Wikipedia, the splitting field L of p over Q is>>>>    Q(cuberoot(3), -1/2 +i*srqrt(3)/2)>>>> where -1/2 +i*srqrt(3)/2 is a non-trivial third root of unity.>>>> By Artin, because L is a splitting field,>> Yes.>>> L is a Galois extension of Q.>> Yes.>>> So L is an abelian extension of Q.>> No, L is not an abelian extension of Q.>>> <http://en.wikipedia.org/wiki/Splitting_field#Cubic_example>.>>>> Then L is an extension of degree 6 (as a vector field over Q)>> of Q.>> Yes.>>> By Galois theory, the automorphisms of L fixing Q form a group>> of order 6.>> Right.>>> By the Kronecker-Weber theorem, L isn't an abelian extension.>> Right, L isn't an abelian extension of Q.>>> But we have a non-abelian group of order 6 ...>> Right.>>> So I guess the automorphism group of L (which fix Q) is>> isomorphic to S_3, the symmetric group on three objects.>>>> So, is this right?>> Yes.>>> Some automorphisms:>> (a) identity>> (b) complex conjugation>>>> Supposedly, there should be 4 more automorphisms of L leaving>> Q invariant.>>>> Perhaps cuberoot(3) can be sent to either of>> cuberoot(3)*(-1/2 +i*srqrt(3)/2),>> cuberoot(3)*(-1/2 -i*srqrt(3)/2)?>> Yes.>>> Anyway, finding and constructing these automorphisms of L>> doesn't look too easy.>> It's not hard.>> For each element of L\Q, an automorphis of L over Q must> send that element of L to one of its conjugates (over Q).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 settingabove,2^(1/3)  is said to be conjugate to2^(1/3) * (-1/2 +i*srqrt(3)/2).I looked at Conjugate (group theory) at Wikipedia here:http://en.wikipedia.org/wiki/Conjugate_%28group_theory%29 .David ?ernie?> To get an automorphism of L over Q,>>     send 2^(1/3)>     to any of the 3 cube roots of 2>>     and send -1/2 +i*srqrt(3)/2>     to any of the 2 nontrivial cube roots of 1>> That yields 6 choices (the above choices are independent),> and for each of those 6 choices there is a uniquely determined> automorphism of L over Q.>>> But I guess people require an extension to be Galois so that the>> fundamental theorem of Galois theory applies ...>> Right, Galois extensions are needed for certain theorems.>> quasi-- 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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