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 \(S_\mathrm{{d}}\), 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 \(S_\mathrm{{d}}\) Nora needs to have at least three moves for each even \(d \ge 4\). For \(d=4\) we show that Nora can always get the Galois group \(S_4\) if Nora starts and they play alternately.
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References
Awtrey, C., Barkley, B., Guinn, J., McCraw, M.: Resolvents, masses, and Galois groups of irreducible quartic polynomials. Pi Mu Epsilon J. 13, 609–618 (2014)
Barai, R.: On determination of Galois group of quartic polynomials. Math. Student 87, 73–85 (2018)
Castillo, A., Dietmann, R.: On Hilbert’s irreducibility theorem. Acta Arith. 180, 1–14 (2017)
Chow, S., Dietmann, R.: Enumerative Galois theory for cubics and quartics. Adv. Math. 372, 107282 (2020)
Chow, S., Dietmann, R.: Towards van der Waerden’s conjecture, preprint at arXiv:2106.14593v1 (2021)
Davenport, H., Lewis, D.H., Schinzel, A.: Polynomials of certain special types. Acta Arith. 9, 107–116 (1964)
Evertse, J.-H.: The number of solutions of decomposable form equations. Invent. Math. 122, 559–601 (1995)
Evertse, J.-H., Schlickewei, H.P., Schmidt, W.M.: Linear equations in variables which lie in a multiplicative group. Ann. of Math. 155(2), 807–836 (2002)
Gasarch, W., Washington, L.C., Zbarsky, S.: The coefficient-choosing game. J. Comb. Number Theory 10, 1–17 (2018)
Hering, H.: Seltenheit der Gleichungen mit Affekt bei linearem Parameter. Math. Ann. 186, 263–270 (1970)
Jacobson, N.: Basic algebra. I. W. H. Freeman and Co., San Francisco, Calif (1974)
Kaplansky, I.: Fields and rings, Chicago Lectures in Mathematics, 2nd edn. University of Chicago Press (1972)
Kappe, L.-C., Warren, B.: An elementary test for the Galois group of a quartic polynomial. Amer. Math. Monthly 96, 133–137 (1989)
Schinzel, A.: Polynomials with special regard to reducibility. Cambridge University Press, Cambridge, UK (2000)
Schlickewei, H.P.: \(S\)-unit equations over number fields. Invent. Math. 102, 95–107 (1990)
Sharma, D., Singhal, L.: On the coefficient-choosing game. Mosc. J. Comb. Number Theory 10, 183–202 (2021)
van der Waerden, B.L.: Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt. Monatsh. Math. Phys. 43, 137–147 (1936)
Viana, P., Veloso, P.M.: Galois theory of reciprocal polynomials. Amer. Math. Monthly 109, 466–471 (2002)
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This research has received funding from European Social Fund (Project No 09.3.3-LMT-K-712-01-0037) under grant agreement with the Research Council of Lithuania (LMTLT).
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Communicated by Rosihan M. Ali.
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Dubickas, A. A Game with Two Players Choosing the Coefficients of a Polynomial. Bull. Malays. Math. Sci. Soc. 45, 793–805 (2022). https://doi.org/10.1007/s40840-021-01219-3
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DOI: https://doi.org/10.1007/s40840-021-01219-3