Minimal pairs, inertia degrees, ramification degrees and implicit constant fields. Issue 11 (19th August 2022)

• Record Type: Journal Article
• Authors: Dutta, Arpan
• Is Part Of: Communications in algebra. Volume 50:Issue 11(2022)
• Page Start: 4964
• Page End: 4974
• Publication Date: 2022-08-19
• Subjects: extensions of valuation to rational function fields -- implicit constant fields -- minimal pairs -- ramification theory -- valuation -- valuation transcendental extensions; 12J20 -- 13A18 -- 12J25; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2022.2078833
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: An extension ( K ( X ) | K, v ) of valued fields is said to be valuation transcendental if we have equality in the Abhyankar inequality. Minimal pairs of definition are fundamental objects in the investigation of valuation transcendental extensions. In this article, we associate a uniquely determined positive integer with a valuation transcendental extension. This integer is defined via a chosen minimal pair of definition, but it is later shown to be independent of the choice. Further, we show that this integer encodes important information regarding the implicit constant field of the extension ( K ( X ) | K, v ) .