Minimal pairs, inertia degrees, ramification degrees and implicit\n constant fields

2021/11/26 by Arpan Dutta, Dutta, Arpan · 1 citation
Mathematics · #12J20 #12J25 #13A18 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #FOS: Mathematics #History and Theory of Mathematics #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2111.13641

openalex publication_date 2021/11/26 · openalex created_date 2022/11/06 · openalex updated_date 2026/07/28

Abstract

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

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