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Cyclic Extensions with Prescribed Ramification

  • Georges Gras
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we give an approach to the study of ramification in \({\bar K^{ab}}\left[ {{p^e}} \right]/K\), the maximal pro-p-subextension of\({\bar K^{ab}}/K\) with exponent p e , in particular through the study of the ramification possibilities for cyclic extensions of degree p e of K. We will apply these results to the case of the maximal tamely ramified abelian extension \(H_{ta}^{res}/K\) whose structure is always complicated as soon as the invariants \({{\rm A}^{res}}\) or E res are nontrivial. Concerning this, we will have to make an assumption on the group \({\left( {{{\rm A}^{res}}} \right)_p}\) when e ≥ 2, but the case e = 1 can be solved without any assumption.

Keywords

Exact Sequence Exceptional Case Quadratic Extension Abelian Extension Cyclic Extension 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Georges Gras
    • 1
  1. 1.Faculty of Sciences, Laboratory of Mathematics and CNRSUniversity of Franche-ComtéBesançon CedexFrance

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