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

CITATION: Marques, S. & Ward, J. 2019. A complete study of the ramification for any separable cubic global function field. Research in Number Theory, 5(36). doi:10.1007/s40993-019-0173-y

The original publication is available at https://www.springer.com/journal/40993

Article

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.