Discriminants of Chebyshev Radical Extensions

2013/04/22 by Thomas Alden Gassert, Gassert, Thomas Alden
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1304.6055

This update contains proofs for the conjectures appearing in a earlier version of this paper. This article draws heavily from arXiv:0906.2629

arxiv created 2013/06/07 · arxiv updated 2013/06/10

Abstract

Let t be any integer and fix an odd prime ell. Let Phi(x) = T_ellⁿ(x)-t denote the n-fold composition of the Chebyshev polynomial of degree ell shifted by t. If this polynomial is irreducible, let K = bbq(theta), where theta is a root of Phi. A theorem of Dedekind's gives a condition on t for which K is monogenic. For other values of t, we apply the Montes algorithm to obtain a formula for the discriminant of K and to compute basis elements for the ring of integers O_K.

Citations

• Newton polygons and p-integral bases

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