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Cubic polynomials defining monogenic fields with the same discriminant. (English. French summary) Zbl 1473.11198

Let \(K\) be a number field with ring of integers \(\mathcal{O}_K\). \(K\) is defined to be monogenic if \(\mathcal{O}_K = \mathbb{Z}[x]\) for some \(x\) in \(\mathcal{O}_K\), where \(\mathbb{Z}\) is the ring of rational integers. In this article, the authors show that there exist infinitely many triples of polynomials defining distinct monogenic cubic fields over the rationals \(\mathbb{Q}\), with the same discriminant. To show that their polynomials are monogenic they use the criteria that the polynomial discriminant is equal to the field discriminant, and to establish this they rely extensively on the tables in a 1998 article by S. Alaca [Proc. Am. Math. Soc. 126, No. 7, 1949–1953 (1998; Zbl 0908.11048)]. They illustrate this in detail for one of their three polynomial forms, and leave it to the reader to verify this for their other two polynomial forms. They subsequently utilize basic field theory to give an indication that their three fields \(K_1\), \(K_2\), \(K_3\), each defined by adjoining a root of the corresponding above-mentioned polynomial, are distinct. They show this in detail for two particular fields and again leave the remaining two cases to the reader. To show that there are infinitely many triplets of these polynomials they utilize a 1995 article by C. L. Stewart and J. Top [J. Am. Math. Soc. 8, No. 4, 943–973 (1995; Zbl 0857.11026)]. Finally, they illustrate their results with a number of examples of triplets of polynomials defining distinct monogenic cubic fields with the same discriminant, and they give one example with four such polynomials.

MSC:

11R16 Cubic and quartic extensions
11R29 Class numbers, class groups, discriminants
Full Text: DOI

References:

[1] Saban Alaca, \(p\)-integral bases of a cubic field, Proc. Am. Math. Soc.126 (1998), p. 1949-1953 · Zbl 0908.11048
[2] Zenon I. Borevich & Igor R. Shafarevich, Number Theory, Academic Press Inc., 1966 · Zbl 0145.04902
[3] Pascual Llorente & Enric Nart, Effective determination of the decomposition of the rational primes in a cubic field, Proc. Am. Math. Soc.87 (1983), p. 579-585 · Zbl 0514.12003
[4] Daniel C. Mayer, “How many fields share a common discriminant? (Multiplicity problem)”, Algebra and Algebraic Number Theory,
[5] Richard A. Mollin, Algebraic Number Theory, Discrete Mathematics and its Applications, CRC Press, 2011 · Zbl 1219.11001
[6] Cameron L. Stewart & Jaap Top, On ranks of twists of elliptic curves and power-free values of binary forms, J. Am. Math. Soc.8 (1995), p. 943-973 · Zbl 0857.11026
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