{"title":"关于有区分子域的代数闭域","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":"{\"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}","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
摘要
本文关注对（K, F）的模型理论研究，其中 K 是一个代数闭域，F 是允许额外结构的 K 的一个区分子域。我们研究这些模型对的基本模型理论性质，如量子消元、模型完备性和饱和模型。我们还证明了分类理论概念的一些保存结果，如稳定性、简单性、NSOP1 和 NIP。作为应用，我们得出结论：如果一个 PAC 域的绝对伽罗瓦群是 NSOP1（作为一个无穷群），那么这个 PAC 域就是 NSOP1。
On algebraically closed fields with a distinguished subfield
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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