Monogenic fields arising from trinomials

Ibarra, Ryan; Lembeck, Henry; Ozaslan, Mohammad; Smith, Hanson; Stange, Katherine E.

doi:10.2140/involve.2022.15.299

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Monogenic fields arising from trinomials

Ryan Ibarra, Henry Lembeck, Mohammad Ozaslan, Hanson Smith and Katherine E. Stange

Vol. 15 (2022), No. 2, 299–317

DOI: 10.2140/involve.2022.15.299

Abstract

We call a polynomial monogenic if a root $𝜃$ has the property that $ℤ [𝜃]$ is the full ring of integers of $ℚ (𝜃)$ . Consider the two families of trinomials $x^{n} + a x + b$ and $x^{n} + c x^{n - 1} + d$ . For any $n > 2$ , we show that these families are monogenic infinitely often and give some positive densities in terms of the coefficients. When $n = 5$ or 6 and when a certain factor of the discriminant is square-free, we use the Montes algorithm to establish necessary and sufficient conditions for monogeneity, illuminating more general criteria given by Jakhar, Khanduja, and Sangwan using other methods. Along the way we remark on the equivalence of certain aspects of the Montes algorithm and Dedekind’s index criterion.

Keywords

monogenic, power integral basis, ring of integers, trinomial

Mathematical Subject Classification

Primary: 11R04

Milestones

Received: 10 May 2021

Accepted: 3 September 2021

Published: 29 July 2022

Communicated by Nathan Kaplan

Authors

Ryan Ibarra
Department of Mathematics University of Colorado Boulder, CO United States

Henry Lembeck
Department of Mathematics University of Colorado Boulder, CO United States

Mohammad Ozaslan
Department of Mathematics University of Colorado Boulder, CO United States

Hanson Smith
Department of Mathematics University of Connecticut Storrs, CT United States

Katherine E. Stange
Department of Mathematics University of Colorado Boulder, CO United States

Vol. 15, No. 2, 2022