Monogenic polynomials with non-squarefree discriminant
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Abstract:
We say a monic polynomial $f(x)\in \mathbb {Z}[x]$ of degree $n$ is monogenic if $f(x)$ is irreducible over $\mathbb {Q}$ and $\left \{1,\theta ,\theta ^2,\ldots ,\theta ^{n-1}\right \}$ is a basis for the ring of integers of $\mathbb {Q}(\theta )$, where $f(\theta )=0$. In 2012, for any integer $n\ge 2$, Kedlaya gave a construction to produce infinitely many monic integer-coefficient irreducible polynomials of degree $n$ having squarefree discriminant. Such polynomials are necessarily monogenic. In this article, for any prime $p\ge 3$, we extend Kedlaya’s methods to construct explicit infinite families of monogenic polynomials of degree $p$ having non-squarefree discriminant.References
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Additional Information
- Lenny Jones
- Affiliation: Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania 17257
- Address at time of publication: 193 Summer Breeze Lane, Chambersburg, Pennsylvania 17202
- MR Author ID: 265349
- Email: lkjone@ship.edu
- Received by editor(s): April 27, 2019
- Received by editor(s) in revised form: August 21, 2019, and August 30, 2019
- Published electronically: November 13, 2019
- Communicated by: Rachel Pries
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1527-1533
- MSC (2010): Primary 11R04; Secondary 11R09
- DOI: https://doi.org/10.1090/proc/14858
- MathSciNet review: 4069191