Artin’s Theorem Towards the Existence of Algebraic Closures

نویسندگان

چکیده

Summary This is the first part of a two-part article formalizing existence and uniqueness algebraic closures using Mizar system [1], [2]. Our proof follows Artin’s classical one as presented by Lang in [3]. In this we prove that for given field F there exists extension E such every non-constant polynomial p ∈ [ X ] has root . applies Kronecker’s construction to each \F simultaneously. To do so need ring 1 , 2 ...] with infinitely many variables, polynomal The desired then \I where I maximal ideal generated all polynomials ]. Note, show Zorn’s lemma be applied. second iterated giving an infinite sequence fields, whose union establishes A which root. elements closure closures, e.g. two are isomorphic over technique extending monomorphisms applied: monomorphism → can extended any case algebraically closed isomorphism. Note again relies on lemma.

برای دانلود باید عضویت طلایی داشته باشید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Power Series and p-adic Algebraic Closures

In a previous paper [3], the author gave an explicit description of the algebraic closure of the power series field over a field of characteristic p > 0, in terms of certain “generalized power series”. The purpose of the present paper is to extend this work to mixed characteristic. Specifically, we give an analogous description of the algebraic closure of the Witt ring W (K) of an algebraically...

متن کامل

Closures in Formal Languages and Kuratowski's Theorem

A famous theorem of Kuratowski states that in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where closure is either Kleene closure or positive closure. We classify languages according to the structure of the algebra they generate under it...

متن کامل

Derived Algebraic Geometry XII: Proper Morphisms, Completions, and the Grothendieck Existence Theorem

1 Generalities on Spectral Algebraic Spaces 5 1.1 Scallop Decompositions in the Separated Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Quasi-Finite Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Quasi-Separatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Points of Spectral Algebr...

متن کامل

a tripartite model of efl teachers attributions, burnout, and ‎self-regulation: towards the prospects of effective teaching

همطالعه حاضر به ارائه مدلی برای آموزش موثر زبان انگلیسی ‏می پردازد. مدل حاضر از سه عامل تاثیر گذار در کارایی ‏تدریس معلمان زبان انگلیسی بهره می برد. این سه عامل شامل ‏سبکهای اسنادی، خود تنطیمی و فرسودگی شغلی معلمان ایرانی ‏زبان انگلیسی می باشد. رساله مورد نظر درچهار فاز طراحی ‏شده است: فاز اول شامل طراحی و رواسازی پرسشنامه سبکهای ‏اسنادی معلمان زبان انگلیسی و فاز دوم شامل استفاده از ‏این پرسشنا...

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ژورنال

عنوان ژورنال: Formalized Mathematics

سال: 2022

ISSN: ['1898-9934', '1426-2630']

DOI: https://doi.org/10.2478/forma-2022-0014