Research in Number Theory

, 5:36 | Cite as

A complete study of the ramification for any separable cubic global function field

  • Sophie MarquesEmail author
  • Jacob Ward


We explicitly describe the ramified places in any separable cubic extension of a cubic global function field in terms of a unique given parameter. This is all done using the uniqueness of the purely cubic closure, which is a useful new tool for the study of cubic function fields. We give a notion of local standard forms, that is useful for many purposes, including classifying and computing of integral bases. We then determine explicitly the genus of any separable cubic extension of any global function field given the minimal polynomial of the extension. The formulae we obtain is particularly useful for further study owing to the well-understood and straightforward close relation between the parameter we define and ramification within the extension.


Cubic Function field Finite field Genus Ramification 

Mathematics Subject Classification

(Primary ): 11T06 (Secondary): 11R32 11R16 11T55 11R58 



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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of StellenboschStellenboschSouth Africa

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