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A Game with Two Players Choosing the Coefficients of a Polynomial

A Game with Two Players Choosing the Coefficients of a Polynomial We consider some versions of a game when two players Nora and Wanda in some order are choosing the coefficients of a degree d polynomial. The aim of Nora is to get a polynomial which has no roots in some field or, more generally, is irreducible over that field or, even more generally, has the largest possible Galois group Sd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_\mathrm{{d}}$$\end{document}, while the aim Wanda is the opposite. We show that in order to obtain an irreducible polynomial for Nora it suffices to have the last move. However, to ensure that the splitting field of the resulting polynomial with integer coefficients has Galois group Sd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_\mathrm{{d}}$$\end{document} Nora needs to have at least three moves for each even d≥4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d \ge 4$$\end{document}. For d=4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d=4$$\end{document} we show that Nora can always get the Galois group S4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_4$$\end{document} if Nora starts and they play alternately. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

A Game with Two Players Choosing the Coefficients of a Polynomial

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Publisher
Springer Journals
Copyright
Copyright © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2021
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-021-01219-3
Publisher site
See Article on Publisher Site

Abstract

We consider some versions of a game when two players Nora and Wanda in some order are choosing the coefficients of a degree d polynomial. The aim of Nora is to get a polynomial which has no roots in some field or, more generally, is irreducible over that field or, even more generally, has the largest possible Galois group Sd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_\mathrm{{d}}$$\end{document}, while the aim Wanda is the opposite. We show that in order to obtain an irreducible polynomial for Nora it suffices to have the last move. However, to ensure that the splitting field of the resulting polynomial with integer coefficients has Galois group Sd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_\mathrm{{d}}$$\end{document} Nora needs to have at least three moves for each even d≥4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d \ge 4$$\end{document}. For d=4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d=4$$\end{document} we show that Nora can always get the Galois group S4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_4$$\end{document} if Nora starts and they play alternately.

Journal

Bulletin of the Malaysian Mathematical Sciences SocietySpringer Journals

Published: Mar 1, 2022

Keywords: Roots of polynomials; Hilbert’s irreducibility theorem; Galois group; 11C08; 11R09; 11S05; 91A46

References