Vol. 15, No. 2, 2022

Download this article
Download this article For screen
For printing
Recent Issues

Volume 15
Issue 2, 185–365
Issue 1, 1–184

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Author Index
Coming Soon
 
Other MSP Journals
Monogenic fields arising from trinomials

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

Vol. 15 (2022), No. 2, 299–317
Abstract

We call a polynomial monogenic if a root 𝜃 has the property that [𝜃] is the full ring of integers of (𝜃). Consider the two families of trinomials xn + ax + b and xn + cxn1 + 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