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about the KroneckerWeber theorem
Posted:
Feb 3, 2013 6:25 PM


The KroneckerWeber 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 nontrivial third root of unity.
By Artin, because L is a splitting field, L is a Galois extension of Q. So L is 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.
By Galois theory, the automorphisms of L fixing Q form a group of order 6. By the KroneckerWeber theorem, L isn't an abelian extension.
But we have a nonabelian group of order 6 ...
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?
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) ?
Anyway, finding and constructing these automorphisms of L doesn't look too easy.
But I guess people require an extension to be Galois so that the fundamental theorem of Galois theory applies ...
dave
 dracut:/# lvm vgcfgrestore File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID 993: sh Please specify a *single* volume group to restore.



