Discriminants of Chebyshev radical extensions

T. Alden Gassert[1]

  • [1] University of Colorado, Boulder Campus Box 395 Boulder, CO, USA 80309-0395

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 3, page 607-633
  • ISSN: 1246-7405

Abstract

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Let t be any integer and fix an odd prime ℓ . Let Φ ( x ) = T ℓ n ( x ) - t denote the n -fold composition of the Chebyshev polynomial of degree ℓ shifted by t . If this polynomial is irreducible, let K = ℚ ( θ ) , where θ is a root of Φ . We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on t that ensure K is monogenic. For other values of t , we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of K and compute an integral basis for the ring of integers 𝒪 K .

How to cite

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Gassert, T. Alden. "Discriminants of Chebyshev radical extensions." Journal de Théorie des Nombres de Bordeaux 26.3 (2014): 607-633. <http://eudml.org/doc/275794>.

@article{Gassert2014,
abstract = {Let $t$ be any integer and fix an odd prime $\ell $. Let $\Phi (x) = T_\ell ^n(x)-t$ denote the $n$-fold composition of the Chebyshev polynomial of degree $\ell $ shifted by $t$. If this polynomial is irreducible, let $K = \mathbb\{Q\}(\theta )$, where $\theta $ is a root of $\Phi $. We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on $t$ that ensure $K$ is monogenic. For other values of $t$, we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of $K$ and compute an integral basis for the ring of integers $\{\cal O\}_K$.},
affiliation = {University of Colorado, Boulder Campus Box 395 Boulder, CO, USA 80309-0395},
author = {Gassert, T. Alden},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {12},
number = {3},
pages = {607-633},
publisher = {Société Arithmétique de Bordeaux},
title = {Discriminants of Chebyshev radical extensions},
url = {http://eudml.org/doc/275794},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Gassert, T. Alden
TI - Discriminants of Chebyshev radical extensions
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/12//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 3
SP - 607
EP - 633
AB - Let $t$ be any integer and fix an odd prime $\ell $. Let $\Phi (x) = T_\ell ^n(x)-t$ denote the $n$-fold composition of the Chebyshev polynomial of degree $\ell $ shifted by $t$. If this polynomial is irreducible, let $K = \mathbb{Q}(\theta )$, where $\theta $ is a root of $\Phi $. We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on $t$ that ensure $K$ is monogenic. For other values of $t$, we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of $K$ and compute an integral basis for the ring of integers ${\cal O}_K$.
LA - eng
UR - http://eudml.org/doc/275794
ER -

References

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