Normal Extensions

Schwarzweller, Christoph

doi:10.2478/forma-2023-0011

Abstract

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(a₁, . . . 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]).

References

Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.
Search in Google Scholar Back to article
Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.
Search in Google Scholar Back to article
Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.
Search in Google Scholar Back to article
Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363–371, 2016. doi:10.15439/2016F520.
Search in Google Scholar Back to article
Serge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition).
Search in Google Scholar Back to article
Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991.
Search in Google Scholar Back to article
Piotr Rudnicki, Christoph Schwarzweller, and Andrzej Trybulec. Commutative algebra in the Mizar system. Journal of Symbolic Computation, 32(1/2):143–169, 2001. doi:10.1006/jsco.2001.0456.
Search in Google Scholar Back to article
Christoph Schwarzweller. Artin’s theorem towards the existence of algebraic closures. Formalized Mathematics, 30(3):199–207, 2022. doi:10.2478/forma-2022-0014.
Search in Google Scholar Back to article
Christoph Schwarzweller. Existence and uniqueness of algebraic closures. Formalized Mathematics, 30(4):281–294, 2022. doi:10.2478/forma-2022-0022.
Search in Google Scholar Back to article
Christoph Schwarzweller. Field extensions and Kronecker’s construction. Formalized Mathematics, 27(3):229–235, 2019. doi:10.2478/forma-2019-0022.
Search in Google Scholar Back to article
Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251–261, 2020. doi:10.2478/forma-2020-0022.
Search in Google Scholar Back to article
Christoph Schwarzweller. Splitting fields. Formalized Mathematics, 29(3):129–139, 2021. doi:10.2478/forma-2021-0013.
Search in Google Scholar Back to article
Christoph Schwarzweller and Sara Burgoa. Splitting fields for the rational polynomials x²−2, x²+x+1, x³−1, and x³−2. Formalized Mathematics, 30(1):23–30, 2022. doi:10.2478/forma-2022-0003.
Search in Google Scholar Back to article
Christoph Schwarzweller and Agnieszka Rowińska-Schwarzweller. Algebraic extensions. Formalized Mathematics, 29(1):39–48, 2021. doi:10.2478/forma-2021-0004.
Search in Google Scholar Back to article

Normal Extensions

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