Discriminants of Chebyshev Radical Extensions

TL;DR

This paper investigates the arithmetic properties of Chebyshev radical extensions, providing criteria for monogenicity and explicit formulas for discriminants and integral bases using advanced algebraic number theory techniques.

Contribution

It introduces new methods to determine when Chebyshev radical extensions are monogenic and computes explicit discriminants and integral bases for these fields.

Findings

Derived a criterion for monogenicity of Chebyshev radical extensions.

Obtained explicit formulas for discriminants of these extensions.

Developed algorithms to compute integral bases using the Montes method.

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 = 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.