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2011 | OriginalPaper | Chapter
Function Fields of One Variable over PAC Fields
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We prove that if
K
is an ample field of cardinality
m
and
E
is a function field of one variable over
K
, then Gal(
E
) is semi-free of rank
m
(Theorem 11.7.1). It follows from Theorem 10.5.4 that if
F
is a finite extension of
E
, or an Abelian extension of
E
, or a proper finite extension of a Galois extension of
E
, or
F
is “contained in a diamond” over
E
, then Gal(
F
) is semi-free.
We apply the latter results to the case where
K
is PAC and
E
=
K
(
x
), where
x
is an indeterminate. We construct a
K
-radical extension
F
of
E
in a diamond over
E
and conclude that
F
is Hilbertian and Gal(
F
) is semi-free and projective (Theorem 11.7.6), so Gal(
F
) is free. In particular, if
K
contains all roots of unity of order not divisible by char(
K
), then Gal(
E
)
ab
is free of rank equal to card(
K
) (Theorem 11.7.6).