Discriminants of Chebyshev radical extensions

Abstract

Let t be any integer and fix an odd prime ℓ. Let Φ(x) = T n ℓ (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. 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 OK .

Dynamics of the a-map over residually finite Dedekind domains and applications

“…We obtain ord(a, p 3 ) = ϕ D (p 3 ) = 2. The ν-series associated with a and n 0 equals n 0 (a) = (N D (p 1 p 2 ), N D (p 1 )) = (6,2). From Theorem 3.6, we have that G(Γ 1+ √ −5, 6 ) = Cyc(1, T (6,2) ) ⊕ Cyc(2, T (6,2) ).…”

“…For instance, iterations of quadratic polynomials over finite fields (motivated in part by some cryptographic applications such as the Pollard-rho factorization algorithm) were studied in [9,13,18]. Dynamic of Chebyshev polynomials of prime degree and its relation with decomposition of primes in certain towers of number fields were studied by A. Gassert in [1] and [2]. Dynamic of Chebyshev polynomials of arbitrary degree was studied in [11].…”

Monogenic Binomial Compositions

“…For any fixed degree n ≥ 2, the density of monogenic 1074 Joshua Harrington and Lenny Jones polynomials is 6/π 2 ≈ .607927 [4]. However, determining infinite families of degree-n monogenic polynomials can be difficult, and much research has been done to locate such families [1,2,5,8,10,12,13,16,17,23,29].…”

Monogenic polynomials with non-squarefree discriminant

“…For more information on general monogenic fields, see [23]. For some recent specific examinations of monogenic fields, see [1,2,4,6,8,11,12,13,19,30]. We see from (1) that K being monogenic is equivalent to the existence of some irreducible polynomial f (x), with f (θ) = 0 and K = Q(θ), such that ∆(f ) = ∆(K).…”