Artin’s Theorem Towards the Existence of Algebraic Closures

IF 1 Q1 MATHEMATICS Formalized Mathematics Pub Date : 2022-10-01 DOI:10.2478/forma-2022-0014
Christoph Schwarzweller
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引用次数: 2

Abstract

Summary This is the first part of a two-part article formalizing existence and uniqueness of algebraic closures using the Mizar system [1], [2]. Our proof follows Artin’s classical one as presented by Lang in [3]. In this first part we prove that for a given field F there exists a field extension E such that every non-constant polynomial p ∈ F [X] has a root in E. Artin’s proof applies Kronecker’s construction to each polynomial p ∈ F [X]\F simultaneously. To do so we need the polynomial ring F [X1, X2, ...] with infinitely many variables, one for each polynomal p ∈ F [X]\F . The desired field extension E then is F [X1, X2, ...]\I, where I is a maximal ideal generated by all non-constant polynomials p ∈ F [X]. Note, that to show that I is maximal Zorn’s lemma has to be applied. In the second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial p ∈ A[X] has a root. The field of algebraic elements of A then is an algebraic closure of F . To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism F → A, where A is an algebraic closure of F can be extended to a monomorphism E → A, where E is any algebraic extension of F . In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn’s lemma.
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代数闭包存在性的Artin定理
本文是使用Mizar系统形式化代数闭包的存在唯一性的两部分文章的第一部分[1],[2]。我们的证明遵循Lang在[3]中提出的Artin的经典证明。在第一部分中,我们证明了对于给定的域F存在一个域扩展E,使得每个非常数多项式p∈F [X]在E中都有根。Artin的证明将Kronecker构造同时应用于每个多项式p∈F [X]\F。为此,我们需要多项式环F [X1, X2,…]]有无穷多个变量,每个多项式p∈F [X]\F对应一个变量。所需的域扩展E则为F [X1, X2,…]\I,其中I是由所有非常多项式p∈F [X]生成的极大理想。注意,为了证明I是最大值,必须应用佐恩引理。第二部分对该构造进行迭代,给出一个无限域序列,其并建立F的域扩展a,其中每个非常多项式p∈a [X]都有一个根。则A的代数元域是F的代数闭包。为了证明代数闭包的唯一性,例如F的两个代数闭包在F上是同构的,应用了扩展单态的技术:一个单态F→a,其中a是F的一个代数闭包,可以推广到一个单态E→a,其中E是F的任意代数扩展。如果E是代数闭的,这个单态是同构的。注意,扩展单态的存在性同样依赖于佐恩引理。
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Formalized Mathematics
Formalized Mathematics MATHEMATICS-
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审稿时长
10 weeks
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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