{"title":"On algebraically closed fields with a distinguished subfield","authors":"Christian d’Elbée, Itay Kaplan, Leor Neuhauser","doi":"10.1007/s11856-024-2621-1","DOIUrl":null,"url":null,"abstract":"<p>This paper is concerned with the model-theoretic study of pairs (<i>K, F</i>) where <i>K</i> is an algebraically closed field and <i>F</i> is a distinguished subfield of <i>K</i> allowing extra structure. We study the basic model-theoretic properties of those pairs, such as quantifier elimination, model-completeness and saturated models. We also prove some preservation results of classification-theoretic notions such as stability, simplicity, NSOP<sub>1</sub>, and NIP. As an application, we conclude that a PAC field is NSOP<sub>1</sub> iff its absolute Galois group is (as a profinite group).</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-024-2621-1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper is concerned with the model-theoretic study of pairs (K, F) where K is an algebraically closed field and F is a distinguished subfield of K allowing extra structure. We study the basic model-theoretic properties of those pairs, such as quantifier elimination, model-completeness and saturated models. We also prove some preservation results of classification-theoretic notions such as stability, simplicity, NSOP1, and NIP. As an application, we conclude that a PAC field is NSOP1 iff its absolute Galois group is (as a profinite group).
期刊介绍：
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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