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By Mr (k: R (r and Qingquan Scheidler


The ramification groups and different of a compositum of Artin-Schreier extensions. (English summary) Int. J. Number Theory 6 (2010), no. 7, 1541–1564.1793-7310 Let K be a function field over a perfect constant field of characteristic p> 0, and let L/K be an elementary abelian extension of degree p n. The authors investigate how the behaviour of a place P of K in the extension L/K is related to its behaviour in the family of intermediate extensions M/K of degree p. In particular, they show how the number of such M in which P splits (respectively, is inert, ramifies) determines the ramification index and relative degree of P in L/K, and they express the different exponent in L/K for places above P in terms of the different exponents in the M/K. The proofs depend on a detailed examination of the case n = 2 via Artin-Schreier theory, including a complete determination of the Hilbert ramification groups, together with an induction to obtain the general case. It seems to the reviewer that alternative proofs could be given using the properties of Herbrand’s upper numbering of the ramification groups. The authors also show how to construct non-cyclic extensions L/K (with infinite perfect constant field) in which a place of K is inert; this cannot occur for finite constant fields. Finally, in the case that the constant field is finite, the authors use their results to recover a relation between the zeta-functions of K, L and the M, first obtained by I. M. Duursma, H. Stichtenoth and C. Voss [in Arithmetic, geometry an

Year: 2013
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