On a Galois group arising from an iterated map. (English) Zbl 1418.11146

Shimakura, Masamitsu

Proc. Japan Acad., Ser. A 94, No. 5, 43-48 (2018).

Summary: We study the irreducibility and the Galois group of the polynomial \(f(a,x)=x^{8}+3ax^{6}+3a^{2}x^{4}+(a^{2}+1)ax^{2}+a^{2}+1\) over \(\mathbb{Q}(a)\) and \(\mathbb{Q}\). This polynomial is a factor of the 4-th dynatomic polynomial for the map \(\sigma(x)=x^{3}+ax\).

MSC:

11R32	Galois theory
12F10	Separable extensions, Galois theory
12F20	Transcendental field extensions

Keywords:

dynatomic polynomial; Galois group

Software:

Magma

References:

[1]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3–4, 235–265. · Zbl 0898.68039
[2]	N. Bruin and M. Stoll, The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math. 13 (2010), 272–306. · Zbl 1278.11069
[3]	T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), no. 1, 653–662. · Zbl 0003.15001
[4]	P. Morton, Arithmetic properties of periodic points of quadratic maps, Acta Arith. 62 (1992), no. 4, 343–372. · Zbl 0767.11016
[5]	P. Morton, Characterizing cyclic cubic extensions by automorphism polynomials, J. Number Theory 49 (1994), no. 2, 183–208. · Zbl 0810.12003
[6]	P. Morton, Arithmetic properties of periodic points of quadratic maps. II, Acta Arith. 87 (1998), no. 2, 89–102. · Zbl 1029.12002
[7]	P. Morton and P. Patel, The Galois theory of periodic points of polynomial maps, Proc. London Math. Soc. (3) 68 (1994), no. 2, 225–263. · Zbl 0792.11043
[8]	J. H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, 241, Springer, New York, 2007. · Zbl 1130.37001