Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

Dubickas, Artūras; Šiurys, Jonas

doi:10.1007/s12044-016-0325-0

Cited by 3 publications

(1 citation statement)

References 9 publications

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“…Other results concerning these polynomials can be found in [9,10,14]. There are some identities involving p r,t (x) and q r,n (x) which helped establish the results found in [2] and [11].…”

Section: Introductionmentioning

confidence: 62%

On the Galois group over $\mathbb Q$ of a truncated binomial expansion

Filaseta

Moy

2018

Colloq. Math.

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Pour tout nombre entier positif n, les extensions binomiales tronquées de (1 + x) n constitutées de tous les termes de degré ≤ r où 1 ≤ r ≤ n − 2 semblent toujoursêtre irréductibles. Pour r fixe et n suffisamment grand, ce résultat est connu. Nous montrons ici que, pour un nombre entier positif fixe r = 6 et n suffisamment grand, le groupe Galois d'un tel polynôme sur les nombres rationnels est le groupe symétrique S r . Pour r = 6, nous montrons que le nombre de n ≤ N exceptionnels pour lesquels le groupe Galois de ce polynôme n'est pas S r est au plus O(log N).ABSTRACT: For positive integers n, the truncated binomial expansions of (1 + x) n which consist of all the terms of degree ≤ r where 1 ≤ r ≤ n − 2 appear always to be irreducible. For fixed r and n sufficiently large, this is known to be the case. We show here that for a fixed positive integer r = 6 and n sufficiently large, the Galois group of such a polynomial over the rationals is the symmetric group S r . For r = 6, we show the number of exceptional n ≤ N for which the Galois group of this polynomial is not S r is at most O(log N).

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“…Other results concerning these polynomials can be found in [9,10,14]. There are some identities involving p r,t (x) and q r,n (x) which helped establish the results found in [2] and [11].…”

Section: Introductionmentioning

confidence: 62%

On the Galois group over $\mathbb Q$ of a truncated binomial expansion

Filaseta

Moy

2018

Colloq. Math.

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Galois groups of n 0 + n 1 X + ⋯ + n 6 X6

Klahn,

Technau

2023

Int. J. Number Theory

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We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many [Formula: see text]. This complements earlier work of Filaseta and Moy, who studied Galois groups of [Formula: see text] for more general pairs [Formula: see text], but had to admit a possibly infinite exceptional set specifically for [Formula: see text] of at most logarithmic growth in [Formula: see text]. The proof rests upon invoking Faltings’ theorem on a suitable fibration of Galois resolvents to show that this exceptional set is, in fact, finite.

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Irreducibility and Galois groups of truncated binomial polynomials

Laishram,

Yadav

2024

Int. J. Number Theory

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For positive integers [Formula: see text], let [Formula: see text] be the truncated binomial expansion of [Formula: see text] consisting of all terms of degree [Formula: see text] It is conjectured that for [Formula: see text], the polynomial [Formula: see text] is irreducible. We confirm this conjecture when [Formula: see text] Also we show for any [Formula: see text] and [Formula: see text], the polynomial [Formula: see text] is irreducible when [Formula: see text] Under the explicit abc-conjecture, for a fixed [Formula: see text], we give an explicit [Formula: see text] depending only on [Formula: see text] such that [Formula: see text], the polynomial [Formula: see text] is irreducible. Further [Formula: see text], the Galois group associated to [Formula: see text] is the symmetric group [Formula: see text]

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Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

Cited by 3 publications

References 9 publications

On the Galois group over $\mathbb Q$ of a truncated binomial expansion

On the Galois group over $\mathbb Q$ of a truncated binomial expansion

Galois groups of n 0 + n 1 X + ⋯ + n 6 X6

Irreducibility and Galois groups of truncated binomial polynomials

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