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[Mathqa]Re: Galois group of the rationals
Posted:
Mar 5, 2001 2:21 PM
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Nick Halloway wrote:
> Is it true that you could have a subgroup of finite index in the Galois
> group of the algebraic numbers over the rationals that isn't the group
> that fixes an intermediate field?
I doubt it. Profinite Galois groups set up a correspondence between their
closed subgroups and intermediate fields. Subgroups of finite index should
be open and closed. I suppose to see that it is useful to quote the lemma
that inside a subgroup H of index n in G there is a normal subgroup N of
index at most n! . WLOG we can assume a normal subgroup, therefore. The
question reduces to whether there are discontinuous homomorphisms from these
profinite groups to finite groups carrying the discrete topology. I think
this is just a matter of chasing back through the definition of an inverse
system.
Charles
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