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[Mathqa]Re: Galois group of the rationals
Posted:
Mar 6, 2001 2:21 PM
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Gaetan Chenevier wrote:
> Is there an example of such a group?
> Yes, take the subfield of C generated by all the square roots of integers,
call it K.
OK, the final sentence of my posting on this was incorrect. Let me try to
understand the claim. If V is the direct sum of countably many copies of
the cyclic group of order 2, we can consider V as a vector space over the
field with two elements. Its dual vector space V* is the cartesian product
of countably many cyclic groups of order two. The double dual V** will be
bigger than V, which is the Pontryagin dual of V*. That's because V* is
uncountable, and, not very constructively, one can choose a basis and assign
values of a linear functional on it. Therefore there are many subgroups of
index 2 in V* that aren't closed in the product topology.
In the original context V* occurs as a Galois group Gal(K/Q), as one checks
by taking the linearly disjoint subfields made by adjoining sqr(-1) and the
sqr(p) for p prime. This Galois group is a quotient of the Galois group of
the algebraic numbers over Q.
Charles
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