Abstract
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).
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© 2011 Springer-Verlag Berlin Heidelberg
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Jarden, M. (2011). Function Fields of One Variable over PAC Fields. In: Algebraic Patching. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15128-6_11
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DOI: https://doi.org/10.1007/978-3-642-15128-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15127-9
Online ISBN: 978-3-642-15128-6
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