Publication Date:
2014-01-18
Description:
In this paper, we use the Merkurjev–Suslin theorem to determine the structure of arithmetically significant Galois modules that arise from Kummer theory. Let K be a field of characteristic different from a prime , n be a positive integer, and suppose that K contains the ( n )th roots of unity. Let L be the maximal n -elementary abelian extension of K , and set G =Gal( L | K ). We consider the G -module J colone L x / n and denote its socle series by J m . We provide a precise condition, in terms of a map to H 3 ( G , Z/ n ), determining which submodules of J m –1 embed in cyclic modules generated by elements of J m ; therefore, this map provides an explicit description of J m and J m / J m –1 . The description of J m / J m –1 is a new non-trivial variant of the classical Hilbert's Theorem 90. The main theorem generalizes a theorem of Adem, Gao, Karaguezian and Minác that deals with the case m = n =2, and also ties in with current trends in minimalistic birational anabelian geometry over essentially arbitrary fields.
Print ISSN:
0024-6093
Electronic ISSN:
1469-2120
Topics:
Mathematics
