Renamings and a Condition-free Formalization of Kronecker’s Construction

In [7], [9], [10] we presented a formalization of Kronecker’s construction of a field extension E for a field F in which a given polynomial p ∈ F [X]\F has a root [5], [6], [3]. A drawback of our formalization was that it works only for polynomial-disjoint fields, that is for fields F with F ∩ F [X] = ∅. The main purpose of Kronecker’s construction is that by induction one gets a field extension of F in which p splits into linear factors. For our formalization this means that the constructed field extension E again has to be polynomial-disjoint.Institute of Informatics, University of Gdansk, PolandGrzegorz 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.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.Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985.Artur Korniłowicz. Quotient rings. Formalized Mathematics, 13(4):573–576, 2005.Heinz Lüneburg. Gruppen, Ringe, Körper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991.Christoph Schwarzweller. On roots of polynomials over F [X]/ 〈 p〉. Formalized Mathematics, 27(2):93–100, 2019. doi:10.2478/forma-2019-0010.Christoph Schwarzweller. On monomorphisms and subfields. Formalized Mathematics, 27(2):133–137, 2019. doi:10.2478/forma-2019-0014.Christoph Schwarzweller. Field extensions and Kronecker’s construction. Formalized Mathematics, 27(3):229–235, 2019. doi:10.2478/forma-2019-0022.Christoph Schwarzweller. Representation matters: An unexpected property of polynomial rings and its consequences for formalizing abstract field theory. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2018 Federated Conference on Computer Science and Information Systems, volume 15 of Annals of Computer Science and Information Systems, pages 67–72. IEEE, 2018. doi:10.15439/2018F88.28212913