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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: 3,250
Registered: 12/13/04
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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