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about the Kronecker-Weber theorem
Posted:
Feb 3, 2013 6:25 PM
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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, 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 Kronecker-Weber theorem,
L isn't an abelian extension.
But we have a non-abelian 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.
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