Algebraic Extensions

In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.Christoph Schwarzweller - Institute of Informatics, University of Gdansk, PolandGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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 Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. 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.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.Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985.Serge Lang. Algebra. Springer, 3rd edition, 2005.Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251–261, 2020. doi:10.2478/forma-2020-0022.291394

Ring and Field Adjunctions, Algebraic Elements and Minimal Polynomials

In [6], [7] we presented a formalization of Kronecker’s construction of a field extension of a field F in which a given polynomial p ∈ F [X]\F has a root [4], [5], [3]. As a consequence for every field F and every polynomial there exists a field extension E of F in which p splits into linear factors. It is well-known that one gets the smallest such field extension – the splitting field of p – by adjoining the roots of p to F. In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomials and prove a number of facts necessary to develop the theory of splitting fields, in particular that for an algebraic element a over F a basis of the vector space F (a) over F is given by a0, . . ., an−1, where n is the degree of the minimal polynomial of a over F .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.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. Renamings and a condition-free formalization of Kronecker’s construction. Formalized Mathematics, 28(2):129–135, 2020. doi:10.2478/forma-2020-0012.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.Yasushige Watase. Algebraic numbers. Formalized Mathematics, 24(4):291–299, 2016. doi:10.1515/forma-2016-0025.28325126

Learning styles: Individualizing computer‐based learning environments

In spite of its importance, learning style is a factor that has been largely ignored in the design of educational software. Two issues concerning a specific set of learning styles, described by Honey and Mumford (1986), are considered here. The first relates to measurement and validity. This is discussed in the context of a longitudinal study to test the predictive validity of the questionnaire items against various measures of academic performance, such as course choice and level of attainment in different subjects. The second issue looks at how the learning styles can be used in computer‐based learning environments. A re‐examination of the four learning styles (Activist, Pragmatist, Reflector and Theorist) suggests that they can usefully be characterized using two orthogonal dimensions. Using a limited number of pedagogical building blocks, this characterization has allowed the development of a teaching strategy suitable for each of the learning styles. Further work is discussed, which will use a multi‐strategy basic algebra tutor to assess the effect of matching teaching strategy to learning style

Auditing scholarly journals published in Malaysia and assessing their visibility

The problem with the identification of Malaysian scholarly journals lies in the lack of a current and complete listing of journals published in Malaysia. As a result, librarians are deprived of a tool that can be used for journal selection and identification of gaps in their serials collection. This study describes the audit carried out on scholarly journals, with the objectives (a) to trace and characterized scholarly journal titles published in Malaysia, and (b) to determine their visibility in international and national indexing databases. A total of 464 titles were traced and their yearly trends, publisher and publishing characteristics, bibliometrics and indexation in national, international and subject-based indexes were described

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

A single journal study : Malaysian Journal of Computer Science

Single journal studies are reviewed and measures used in the studies are highlighted. The following quantitative measures are used to study 272 articles published in Malaysian Journal of Computer Science, (1) the article productivity of the journal from 1985 to 2007, (2) the observed and expected authorship productivity tested using Lotka's Law of author productivity, identification and listing of core authors; (3) the authorship, co-authorship pattern by authors' country of origin and institutional affiliations; (4) the subject areas of research; (5) the citation analysis of resources referenced as well as the age and half-life of citations; the journals referenced and tested for zonal distribution using Bradford's law of journal scattering; the extent of web citations; and (6) the citations received by articles published in MJCS and impact factor of the journal based on information obtained from Google Scholar, the level of author and journal self-citation