Analogue of the Hyodo Inequality for the Ramification Depth in Degree p2 Extensions

Abstract

Ramification in complete discrete valuation fields is studied. For the case of a perfect residue field, there is a well-developed theory of ramification groups. Hyodo introduced the concept of ramification depth associated with the different of an extension and obtained an inequality that combines the concept of ramification depth in a degree p² cyclotomic extension with the concept of ramification depth in a degree p subextension. The paper gives a detailed consideration of the structure of degree p² extensions that can be obtained by a composite of two degree p extensions. In this case, it is not required that the residue field be perfect. Using the concepts of wild and ferocious extensions and the defect of the main unit, degree p² extensions are classified and more accurate estimates for the ramification depth are obtained. In a number of cases, exact formulas for ramification depth are presented.