On absolute Galois splitting fields of central simple algebras.pdf

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On absolute Galois splitting fields of central simple algebras.pdf

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On absolute Galois splitting fields of central simple algebras

a r X i v : m a t h / 0 7 0 2 6 9 3 v 1 [ m a t h .N T ] 2 3 F e b 2 0 0 7 ON ABSOLUTE GALOIS SPLITTING FIELDS OF CENTRAL SIMPLE ALGEBRAS TIMO HANKE1 Abstract. A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper provides an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. A closely related statement holds for division algebras over Henselian valued fields with global residue field. The existence of absolute Galois splitting fields in central simple alge- bras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang Theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit exam- ples of noncrossed product twisted Laurent series rings can not be avoided by starting the construction with a symbol algebra. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Number Theory. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version will be subsequently published in Journal of Number Theory (2007), doi:10.1016/j.jnt.2006.12.011. Introduction A global field is a finite extension of Q or Fp(t) and by a valuation on it we mean a nontrivial absolute value in the nonnegative real numbers, archimedian or non- archimedian. Let L/K be an extension of global