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The absolute Galois group of a
PRC ( = pseudo real closed) field is characterized as a real projective group.
Specifically, it is known that if E is a PRC field, then its absolute Galois group G(E)
is real projective. Conversely, if G is a real projective group, then there
exists a PRC field E such that G(E)≅G. The construction of E makes it of
infinite transcendence degree over Q. However, if a field E is algebraic over Q,
then rank G(E) ≤ℵ0. Therefore it is natural to ask whether for a given
real projective group G of rank ≤ℵ0 we may choose E to be algebraic over
Q.
There are two reasons for asking this question. First of all, the corresponding
question for projective groups and PAC fields is known to have an affirmative answer,
since there exist algebraic PAC fields E such that G(E)≅Fω = the free profinite
group of ranks ℵ0 and since every projective group G of rank ≤ℵ0 is isomorphic to a
closed subgroup of Fω. A generalization of this fact to real projective groups and
PRC fields will be a contribution to the desired description of the closed subgroups of
G(Q). Secondly, an affirmative answer to this question will give us a necessary tool to
the study of the elementary theory of all PRC fields which are algebraic over
Q.
The main goal of this work is indeed to give the desired affirmative answer:
Theorem. If K is a countable formally real Hilbertian field and G is a real projective
group of rank ≤ℵ0, then there exists a PRC algebraic extension E of K such that
G(K)≅G.
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