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Nguyen Duy Tan

Publications and source records attributed to Nguyen Duy Tan.

At least 19 recordsLinked to original sources
arXiv2026

Cyclotomic factors of rational necklace functions

Necklace polynomials arise in various fields of mathematics, including combinatorics, Lie theory, and Galois theory over finite fields. Their arithmetic properties have been extensively studied in recent years. In this article, we introduce a new class of rational necklace functions that unifies two well-studied families of polynomials: necklace polynomials and Fekete polynomials. We describe several ways in which cyclotomic polynomials appear as factors of these rational necklace functions. Our results unify and generalize various earlier work on necklace polynomials and on Fekete polynomials. In particular, we describe a surprising phenomenon in which certain Galois groups play a hidden role in the appearance of new cyclotomic factors that are not covered by these previous works.

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arXiv2025

Zassenhaus filtrations as intersections

Zassenhaus filtrations of profinite groups are an important tool to study profinite groups.In this paper, we describe Zassenhaus filtrations of profinite groups as intersections of kernels of certain representations. In this way we introduce a link between studying profinite groups with methods of Zassenhaus filtrations and representation theory.

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arXiv2024

On the strong Massey property for number fields

Let $n\geq 3$. We show that for every number field $K$ with $ζ_p \notin K$, the absolute and tame Galois groups of $K$ satisfy the strong $n$-fold Massey property relative to $p$. Our work is based on an adapted version of the proof of the Theorem of Scholz-Reichardt.

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arXiv2023

Quaternion algebras and square power classes over biquadratic extensions

Recently the Galois module structure of square power classes of a field $K$ has been computed under the action of $\text{Gal}(K/F)$ in the case where $\text{Gal}(K/F)$ is the Klein $4$-group. Despite the fact that the modular representation theory over this group ring includes an infinite number of non-isomorphic indecomposable types, the decomposition for square power classes includes at most $9$ distinct summand types. In this paper we determine the multiplicity of each summand type in terms of a particular subspace of $\text{Br}(F)$, and show that all "unexceptional" summand types are possible.

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arXiv2023

Galois module structure of some elementary $p$-abelian extensions

We determine the Galois module structure of the parameterizing space of elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the absolute Galois group of $F$ is a free, finitely generated pro-$p$ group, and that $F$ contains a primitive $p$th root of unity if $\text{char}(F) \neq p$.

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arXiv2022

Fekete polynomials, quadratic residues, and arithmetic

Fekete polynomials associate with each prime number $p$ a polynomial with coefficients $-1$ or $1$ except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo $p$. These polynomials were first considered in the 19th century in relation to the studies of Dirichlet $L$-functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomials. In particular, we propose a conjecture on the structure of these Galois groups.

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arXiv2020

Koszul algebras and quadratic duals in Galois cohomology

We investigate the Galois cohomology of finitely generated maximal pro-$p$ quotients of absolute Galois groups. Assuming the well-known conjectural description of these groups, we show that Galois cohomology has the PBW property. Hence in particular it is a Koszul algebra. This answers positively a conjecture by Positselski in this case. We also provide an analogous unconditional result about Pythagorean fields. Moreover, we establish some results that relate the quadratic dual of Galois cohomology with $p$-Zassenhaus filtration on the group. This paper also contains a survey of Koszul property in Galois cohomology and its relation with absolute Galois groups.

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arXiv2020

Enhanced Koszul properties in Galois cohomology

We prove that Galois cohomology satisfies several surprisingly strong versions of Koszul properties, under a well known conjecture, in the finitely generated case. In fact, these versions of Koszulity hold for all finitely generated maximal pro-$p$ quotients of absolute Galois groups which are currently understood. We point out several of these unconditional results which follow from our work. We show how these enhanced versions are preserved under certain natural operations on algebras, generalising several results that were previously established only in the commutative case.

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arXiv2019

Relations in the maximal pro-$p$ quotients of absolute Galois groups

We observe that some basic but fundamental constructions in Galois theory can be used to obtain some interesting restrictions on the structure of Galois groups of maximal $p$-extensions of fields containing a primitive $p$th root of unity. This is an extension of some significant ideas of Demushkin, Labute and Serre from local fields to all fields containing a primitive $p$th root of unity. Our techniques use certain natural simple Galois extensions together with some considerations in Galois cohomology and Massey products.

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arXiv2016

Counting Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions using Massey products

We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed Galois group ${\mathbb U}_4({\mathbb F}_p)$ consisting of unipotent four by four matrices over ${\mathbb F}_p$. Further applications of this method involve the counting of certain Galois extensions with restricted ramifications, and counting the numbers of Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions of some other fields. For each Demushkin pro-$p$-group, we find a very simple version of the condition when the $n$-fold Massey product of one-dimensional cohomological elements of $G$ with coefficients in ${\mathbb F}_p$, is defined. As an easy consequence, we determine those ${\mathbb U}_n({\mathbb F}_p)$ which occur as an epimorphic image of any given Demushkin group.

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arXiv2016

Construction of unipotent Galois extensions and Massey products

For all primes $p$ and all fields, we find a sufficient and necessary condition of the existence of a unipotent Galois extension of degree $p^6$. The main goal of this paper is to describe an explicit construction of such a Galois extension over fields admitting such a Galois extension. This construction is surprising in its simplicity and generality. The problem of finding such a construction has been left open since 2003. Recently a possible solution of this problem gained urgency because of an effort to extend new advances in Galois theory and its relations with Massey products in Galois cohomology.

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arXiv2016

Description of Galois unipotent extensions

Given an arbitrary field $F$, we describe all Galois extensions $L/F$ whose Galois groups are isomorphic to the group of upper triangular unipotent 4-by-4 matrices with entries in the field of two elements.

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arXiv2016

Triple Massey products vanish over all fields

We show that the absolute Galois group of any field has the vanishing triple Massey product property. Several corollaries for the structure of maximal pro-$p$-quotient of absolute Galois groups are deduced. Furthermore, the vanishing of some higher Massey products is proved.

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arXiv2015

Triple Massey products over global fields

Let $K$ be a global field which contains a primitive $p$-th root of unity, where $p$ is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for $p=2$, any triple Massey product over $K$ with respect to $\mathbb{F}_p$, contains 0 whenever it is defined. We show that this is true for all primes $p$.

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arXiv2015

Dimensions of Zassenhaus filtration subquotients of some pro-$p$-groups

We compute the ${\mathbb F}_p$-dimension of an $n$-th graded piece $G_{(n)}/G_{(n+1)}$ of the Zassenhaus filtration for various finitely generated pro-$p$-groups $G$. These groups include finitely generated free pro-$p$-groups, Demushkin pro-$p$-groups and their free pro-$p$ products. We provide a unifying principle for deriving these dimensions.

math.GR↗
arXiv2014

The Kernel Unipotent Conjecture and the vanishing of Massey products for odd rigid fields

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel $n$-Unipotent Conjecture and the Vanishing $n$-Massey Conjecture for $n\geq 3$ were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of $p$-maximal extensions are free pro-$p$-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish a number of further related results.

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arXiv2014

Triple Massey products and Galois theory

We show that any triple Massey product with respect to prime 2 contains 0 whenever it is defined over any field. This extends the theorem of M. J. Hopkins and K. G. Wickelgren, from global fields to any fields. This is the first time when the vanishing of any $n$-Massey product for some prime $p$ has been established for all fields. This leads to a strong restriction on the shape of relations in the maximal pro-2-quotients of absolute Galois groups, which was out of reach until now. We also develop an extension of Serre's transgression method to detect triple commutators in relations of pro-$p$-groups, where we do not require that all cup products vanish. We prove that all $n$-Massey products, $n\geq 3$, vanish for general Demushkin groups. We formulate and provide evidence for two conjectures related to the structure of absolute Galois groups of fields. In each case when these conjectures can be verified, they have some interesting concrete Galois theoretic consequences. They are also related to the Bloch-Kato conjecture.

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