{"title":"Normal Extensions","authors":"Christoph Schwarzweller","doi":"10.2478/forma-2023-0011","DOIUrl":null,"url":null,"abstract":"Summary In this article we continue the formalization of field theory in Mizar [1], [2], [4], [3]. We introduce normal extensions: an (algebraic) extension E of F is normal if every polynomial of F that has a root in E already splits in E . We proved characterizations (for finite extensions) by minimal polynomials [7], splitting fields, and fixing monomorphisms [6], [5]. This required extending results from [11] and [12], in particular that F [ T ] = { p ( a 1 , . . . a n ) | p ∈ F [ X ], a i ∈ T } and F ( T ) = F [ T ] for finite algebraic T ⊆ E . We also provided the counterexample that 𝒬(∛2) is not normal over 𝒬 (compare [13]).","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"15 1","pages":"0"},"PeriodicalIF":1.0000,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Formalized Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/forma-2023-0011","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Summary In this article we continue the formalization of field theory in Mizar [1], [2], [4], [3]. We introduce normal extensions: an (algebraic) extension E of F is normal if every polynomial of F that has a root in E already splits in E . We proved characterizations (for finite extensions) by minimal polynomials [7], splitting fields, and fixing monomorphisms [6], [5]. This required extending results from [11] and [12], in particular that F [ T ] = { p ( a 1 , . . . a n ) | p ∈ F [ X ], a i ∈ T } and F ( T ) = F [ T ] for finite algebraic T ⊆ E . We also provided the counterexample that 𝒬(∛2) is not normal over 𝒬 (compare [13]).
在本文中，我们继续Mizar[1]，[2]，[4]，[3]中场论的形式化。我们引入正规扩展:如果F的每个多项式在E中有一个根已经在E中分裂，那么F的(代数)扩展E是正规的。我们通过极小多项式[7]、分裂域和固定单态[6]、[5]证明了表征(有限扩展)。这需要扩展[11]和[12]的结果，特别是F [T] = {p (a 1，…)。an) | p∈F [X]， ai∈T}， F (T) = F [T]我们还提供了反例，𝒬(∛2)在𝒬上不正常(比较[13])。
期刊介绍：
Formalized Mathematics is to be issued quarterly and publishes papers which are abstracts of Mizar articles contributed to the Mizar Mathematical Library (MML) - the basis of a knowledge management system for mathematics.
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