Infinite families of monogenic trinomials and their Galois groups

Jones, Lenny; Phillips, Tristan

doi:10.1142/s0129167x18500398

Cited by 30 publications

(23 citation statements)

References 39 publications

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“…Theorems 1, 2, and 3 differ from many previous examinations of specific trinomial forms in the literature in that the discriminants of the trinomials here are not squarefree, and their Galois groups can be relatively small; see the upcoming Proposition 1 and following remark. Recently in [20], families of monogenic trinomials have been examined where the discriminant is not squarefree. However, for the trinomials in [20], the Galois group is either the symmetric group S n or the alternating group A n , where n is the degree of the trinomial.…”

Section: Resultsmentioning

confidence: 99%

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Monogenic polynomials with non-squarefree discriminant

Jones

2019

Proc. Amer. Math. Soc.

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For each integer n ≥ 2, we identify new infinite families of monogenic trinomials f (x) = x n + Ax m + B with non-squarefree discriminant, many of which have small Galois group. These families are thus different from many previous examinations of specific trinomial forms in the literature. Moreover, in certain situations when A = B ≥ 2 with fixed n and m, we produce asymptotics on the number of such trinomials with A ≤ X.

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Section: Resultsmentioning

confidence: 99%

“…Recently in [20], families of monogenic trinomials have been examined where the discriminant is not squarefree. However, for the trinomials in [20], the Galois group is either the symmetric group S n or the alternating group A n , where n is the degree of the trinomial. Proposition 1.…”

Section: Resultsmentioning

confidence: 99%

Monogenic polynomials with non-squarefree discriminant

Jones

2019

Proc. Amer. Math. Soc.

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“…Gassert [16] shows that all fields obtained by adjoining a root of x n −a, where a is square-free and a p ≡ a modulo p 2 for all primes p | n, are monogenic. In [27], Jones and Phillips identify infinitely many monogenic fields coming from polynomials of the shape x n +a(m, n)x+b(m, n), where a(m, n) and b(m, n) are prescribed forms. They consider two families of forms, one yielding Galois group S n and the other A n .…”

Section: Previous Workmentioning

confidence: 99%

Two families of monogenic $S_4$ quartic number fields

Smith

2018

Acta Arith.

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Consider the integral polynomials f a,b (x) = x 4 +ax +b and g c,d (x) = x 4 + cx 3 + d. Suppose f a,b (x) and g c,d (x) are irreducible, b | a, and the integers b, d, 256d − 27c 4 , and 256b 3 − 27a 4 gcd(256b 3 , 27a 4 ) are all square-free. Using the Montes algorithm, we show that a root of f a,b (x) or g c,d (x) defines a monogenic extension of Q and serves as a generator for a power basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generating monogenic S4 fields within the families f b,b (x) and g 1,d (x). Corollary 1.1. Consider f b,b (x) = x 4 + bx + b with b ∈ Z and let θ be a root. Suppose b and 256 − 27b are square-free and b = 3, 5. Then, Q(θ) is a monogenic S 4 quartic field and θ is a generator of a power integral basis. Further, at least 29.18% of b ∈ Z satisfy these conditions. Corollary 1.2. Consider g 1,d (x) = x 4 + x 3 + d with d ∈ Z and let τ be a root. Suppose d and 256d − 27 are square-free and d = −2. Then Q(τ ) is a monogenic S 4

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“…The polynomials T (x) in Theorem 1.4 are trinomials and much research has been conducted concerning the mongeneity of trinomials (see [24] and the references therein). Although necessary and sufficient conditions for a trinomial to be monogenic have been given in [22], Theorem 1.4 gives easier and more straightforward conditions to check the monogeneity of the particular trinomials in Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%