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Nicholas Phat Nguyen

Publications and source records attributed to Nicholas Phat Nguyen.

15 recordsLinked to original sources
arXiv2024

A Geometric Representation

This article provides a geometric representation for the well-known isomorphism between the special orthogonal group of an isotropic quadratic space of dimension 3 and the group of projective transformations of a projective line. This geometric representation depends on the theory of inversive transformations in dimension 1 as outlined in the 2021 article Projective Line Revisited by the same author. This geometric representation also provides a new perspective on some classical properties of the projective line, such as the classical cross ratio.

math.HO↗
arXiv2022

A Note on Cyclotomic Polynomials

We present a more general proof that cyclotomic polynomials are irreducible over Q and other number fields that meet certain conditions. The proof provides a new perspective that ties together well-known results, as well as some new consequences, including a necessary condition for the algebraic solution by radicals of certain irreducible polynomials.

math.AC↗
arXiv2020

Valuation and Divisibility

In this paper, we explain how some basic facts about valuation can help clarify many questions about divisibility in integral domains.

math.AC↗
arXiv2020

Four Points and a Quadric

This paper provides a new simple proof of Hesse's theorem in projective geometry for any dimension.

math.HO↗
arXiv2019

The Triangle Altitudes Theorem in Hyperbolic Plane Geometry

We provide a new formulation and proof of the triangle altitudes theorem in hyperbolic plane geometry, together with an easily computed discriminant to distinguish between different basic configurations of the altitudes of such a triangle.

math.HO↗
arXiv2019

Projective Line Revisited

This article provides a new perspective on the geometry of a projective line, which helps clarify and illuminate some classical results about projective plane. As part of the same train of ideas, the article also provides a proof of the nine-point circle theorem valid for any affine plane over any field of characteristic different from 2.

math.GM↗
arXiv2018

An Algebraic Construction of Hyperbolic Planes over a Euclidean Ordered Field

Using concepts and techniques of bilinear algebra, we construct hyperbolic planes over a euclidean ordered field that satisfy all the Hilbert axioms of incidence, order and congruence for a basic plane geometry, but for which the hyperbolic version of the parallel axiom holds rather than the classical Euclidean parallel postulate.

math.HO↗
arXiv2018

A Note on Cyclotomic Integers

In this note, we present a new proof that the cyclotomic integers constitute the full ring of integers in the cyclotomic field.

math.AC↗
arXiv2017

A Note on Graded Rings and Modules

In this note, we consider a situation that is generally used as an intermediate technical step in proving the Artin-Rees lemma but otherwise is not much discussed in introductory accounts of commutative algebra. I hope to show in this note that such technical step deserves more recognition and emphasis in any introduction to commutative algebra because it can be used to prove some significant results in a straight-forward manner, including a generalization of a theorem by Pierre Cartier and John Tate.

math.AC↗