REPOZYTORIUM UNIWERSYTETU
W BIAŁYMSTOKU

Proszę używać tego identyfikatora do cytowań lub wstaw link do tej pozycji: `http://hdl.handle.net/11320/8132`
Pole DCWartośćJęzyk
dc.contributor.authorSchwarzweller, Christoph-
dc.date.accessioned2019-07-29T08:28:48Z-
dc.date.available2019-07-29T08:28:48Z-
dc.date.issued2019-
dc.identifier.citationFormalized Mathematics, Volume 27, Issue 2, Pages 133 - 137-
dc.identifier.issn1426-2630-
dc.identifier.urihttp://hdl.handle.net/11320/8132-
dc.description.abstractThis is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/ as the desired field extension E [5], [3], [4].In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/. Consequently, we translate p along the canonical monomorphism ϕ : F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/.Because F is not a subfield of F [X]/ we construct in this second part the field (E \ ϕF )∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅.Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈 [X] = ∅ and 𝕉 ∩ 𝕉 [X] = ∅, respectively.In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ : F → F [X]/. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root.-
dc.language.isoen-
dc.publisherDeGruyter Open-
dc.subjectroots of polynomials-
dc.subjectfield extensions-
dc.subjectKronecker’s construction-
dc.subject12E05-
dc.subject12F05-
dc.subject68T99-
dc.subject03B35-
dc.titleOn Monomorphisms and Subfields-
dc.typeArticle-
dc.identifier.doi10.2478/forma-2019-0014-
dc.description.AffiliationInstitute of Informatics, University of Gdańsk, Poland-
dc.description.referencesGrzegorz 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.-
dc.description.referencesAdam 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.-
dc.description.referencesNathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985.-
dc.description.referencesHeinz Lüneburg. Gruppen, Ringe, Körper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.-
dc.description.referencesKnut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991.-
dc.identifier.eissn1898-9934-
dc.description.volume27-
dc.description.issue2-
dc.description.firstpage133-
dc.description.lastpage137-
dc.identifier.citation2Formalized Mathematics-
dc.identifier.orcid0000-0001-9587-8737-
Występuje w kolekcji(ach):Formalized Mathematics, 2019, Volume 27, Issue 2

Pliki w tej pozycji:
Plik Opis RozmiarFormat