Adam Chapman

Linkage and Essential $p$ -Dimension

Adam Chapman

We prove that two cyclically linked $p$ -algebras of prime degree become inseparably linked under a prime to $p$ extension if and only if the essential $p$ -dimension of the pair is…

math.RA2023

Biquaternion Algebras, Chain Lemma and Symbol Length

Adam Chapman, Kelly McKinnie

In this note, we present a chain lemma for biquaternion algebras over fields of characteristic 2 in the style of the equivalent chain lemma by Sivatski in characteristic not 2, and…

Common subfields of p-algebras of prime degree

Adam Chapman

We show that if two division $p$ -algebras of prime degree share an inseparable field extension of the center then they also share a cyclic separable one. We show that the converse…

Kummer Subspaces of Tensor Products of Cyclic Algebras

Adam Chapman

We discuss the Kummer subspaces of tensor products of cyclic algebras, focusing mainly on the case of cyclic algebras of degree 3. We present a family of maximal spaces in the gene…

Roots and Critical Points of Polynomials over Cayley--Dickson Algebras

Adam Chapman, Alexander Guterman, Solomon Vishkautsan +1

We study the roots of polynomials over Cayley--Dickson algebras over an arbitrary field and of arbitrary dimension. For this purpose we generalize the concept of spherical roots fr…

Spaces with Vanishing Characteristic Coefficients

Adam Chapman

We prove that the maximal dimension of a subspace $V$ of the generic tensor product of $m$ symbol algebras of prime degree $p$ with $Tr (v^{p - 1}) = 0$ for all $v \in V$ …

Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras

Adam Chapman, Solomon Vishkautsan

Over a composition algebra $A$ , a polynomial $f (x) \in A [x]$ has a root $I^\pm$ if and only $f (x) = g (x) \cdot (x - I^\pm)$ for some $g (x) \in A [x]$ . We examine whether this is true for gener…

cs.LO2026

Reasoning about Medical Triage Optimization with Logic Programming

Jaikrishna Manojkumar Patil, Adam Chapman, Richard Knuszka +2

We present a logic programming framework that orchestrates multiple variants of an optimization problem and reasons about their results to support high-stakes medical decision-maki…

Recurrence relations over division algebras

Adam Chapman, Solomon Vishkautsan

We generalize the solution of linear recurrence relations from fields to central division algebras, adapting the standard tools of companion matrices and characteristic polynomials…

math.RA2013

Chain Lemma for Biquaternion Algebras in Characteristic 2

Adam Chapman

In this paper, we prove that for a given biquaternion algebra over a field of characteristic two, one can move from one symbol presentation to another by at most three steps, such…

math.NT2022

Minimal quadratic forms for the function field of a conic in characteristic $2$

Adam Chapman, Anne QuÃ©guiner-Mathieu

In this note, we construct explicit examples of $F_{Q}$ -minimal quadratic forms of dimension $5$ and $7$ , where $F_{Q}$ is the function field of a conic over a field $F$ of characteris…

math.RA2017

Differential Forms, Linked Fields and the $u$ -Invariant

Adam Chapman, Andrew Dolphin

We associate an Albert form to any pair of cyclic algebras of prime degree $p$ over a field $F$ with $char (F) = p$ which coincides with the classical Albert form when…

General Polynomials and Eigenvalues Over Cayley--Dickson Algebras

Adam Chapman, Ilan Levin, Solomon Vishkautsan

In this paper, we provide an explicit method for determining the zero set of monic quadratic general polynomials over Cayley--Dickson algebras with any base field of characteristic…

The Cyclicity of Tensor Products of Cyclic $p$ -Algebras

Adam Chapman

We revisit the famous theorem of Albert's on the cyclicity of tensor products of cyclic $p$ -algebras. In the case of tensor products of cyclic $p$ -algebras of prime degree, we prov…

Total linkage of quaternion algebras and Pfister forms in characteristic two

Adam Chapman, Andrew Dolphin, Ahmed Laghribi

We study the subfields of quaternion algebras that are quadratic extensions of their center in characteristic 2. We provide examples of the following: two non-isomorphic quaternion…

math.RA2020

Essential Dimension, Symbol Length and $p$ -rank

Adam Chapman, Kelly McKinnie

We prove that the essential dimension of central simple algebras of degree $p^{ℓ m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at l…

On the geometry of zero sets of central quaternionic polynomials II

Gil Alon, Adam Chapman, Elad Paran

Following the work of the first and last authors [2], we further analyze the structure of a zero set of a left ideal in the ring of central polynomials over the quaternion algebra…

Chain Equivalences for Symplectic Bases, Quadratic Forms and Tensor Products of Quaternion Algebras

Adam Chapman

We present a set of generators for the symplectic group which is different from the well-known set of transvections, from which the chain equivalence for quadratic forms in charact…

Kummer Elements in Cyclic Algebras of Degree 5

Adam Chapman

We construct a graph of Kummer elements in a given cyclic algebra of prime degree and study its properties. In case of degree 5, we provide sufficient conditions for two elements t…

math.RA2023

Alternating Roots of Polynomials over Cayley-Dickson Algebras

Adam Chapman, Ilan Levin

We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a gi…

math.AC2019

The descent of biquaternion algebras in characteristic two

Demba Barry, Adam Chapman, Ahmed Laghribi

In this paper we associate an invariant to a biquaternion algebra $B$ over a field $K$ with a subfield $F$ such that $K / F$ is a quadratic separable extension and $\operatorname{cha…

math.RA2011

Clifford algebras of $p$ -central sets

Adam Chapman

A generalization of the term "generalized Clifford algebras" (as appears in papers on advances in applied Clifford algebras) is introduced. This algebra is studied by means of stru…

math.RA2017

Linkage of Quadratic Pfister Forms

Adam Chapman, Shira Gilat, Uzi Vishne

We study the necessary conditions for sets of quadratic $n$ -fold Pfister forms to have a common $(n - 1)$ -fold Pfister factor. For any set $S$ of $n$ -fold Pfister forms generating a…

math.RA2013

Pure Imaginary Roots of Quaternion Standard Polynomials

Adam Chapman

In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equati…

Essential Dimension of Central Simple Algebras of Degree 8 and Exponent 2 in Characteristic 2

Adam Chapman

The goal of this note is to reduce the existing upper bound for the essential dimension of central simple algebras of degree 8 and exponent 2 over fields of characteristic 2 from 1…

Linkage of Sets of Cyclic Algebras

Adam Chapman

Let $p$ be a prime integer and $F$ the function field in two algebraically independent variables over a smaller field $F_{0}$ . We prove that if $char (F_{0}) = p \geq 3$ the…

math.AC2018

Types of Linkage of Quadratic Pfister Forms

Adam Chapman, Andrew Dolphin

Given a field $F$ of positive characteristic $p$ , $I^¸ \in H_{p}^{n - 1} (F)$ and $I^^{2}, I^^{3} \in F^{\times}$ , we prove that if the symbols $I^¸ \land \frac{d I ^ ^{2}}{I ^}^{2}$ and $I^¸ \land \frac{d I ^ ^{3}}{I ^}^{3}$ i…

Clifford semialgebras

Adam Chapman, Letterio Gatto, Louis Rowen

We introduce a theory of Clifford semialgebra systems, with application to representation theory via Hasse-Schmidt derivations on exterior semialgebras. Our main result, after the…

math.RA2019

Linkage of Symbol $p$ -Algebras of Degree 3

Adam Chapman

Given a field $F$ of characteristic $3$ and division symbol $p$ -algebras $[I^\pm, I^^{2})_{3, F}$ and $[I^\pm, I^^{3})_{3, F}$ of degree $3$ over $F$ , we prove that if $Î±\text{dlog}(Î²)\wedge \text…

math.RA2019

The Alternative Clifford Algebra of a Ternary Quadratic Form

Adam Chapman, Uzi Vishne

We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion algebra over the ring of polynomials in one variable over the field of defin…

Cyclic Division Algebras of Odd Prime Degree are never Amitsur-Small

Adam Chapman, Ilan Levin, Marco Zaninelli

A division ring $D$ is Amitsur-Small if for every $n$ and every maximal left ideal $I$ in $D [x_{1}, \dots, x_{n}]$ , $I \cap D [x_{1}, \dots, x_{n - 1}]$ is maximal in $D [x_{1}, \dots, x_{n - 1}]$ . Th…

Roots and Dynamics of Octonion Polynomials

Adam Chapman, Solomon Vishkautsan

This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polyn…

math.NT2025

Essential dimension of sequences of quadratic Pfister forms

Fatma Kader BingÃ¶l, Adam Chapman, Ahmed Laghribi

We study the essential dimension of the set of isometry classes of $m$ -tuples $(I ¨ _{1}, ..., I ¨ _{m})$ of quadratic $n$ -fold Pfister forms over a field $F$ such that the Witt class of $Ï…

math.KT2024

Invariant for Sets of Pfister Forms

Adam Chapman, Ilan Levin

We associate an $(n_{1} + \dots + n_{t} - k (t - 1))$ -fold Pfister form to any $t$ -tuple of $k$ -linked Pfister forms of dimensions $2^{n_{1}}, \dots, 2^{n_{t}}$ , and prove its invariance under the di…

math.RA2020

Factoring Octonion Polynomials

Adam Chapman

We provide an analogue of Wedderburn's factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring…

math.AC2017

The $u^{n}$ -invariant and the Symbol Length of $H_{2}^{n} (F)$

Adam Chapman, Kelly McKinnie

Given a field $F$ of $char (F) = 2$ , we define $u^{n} (F)$ to be the maximal dimension of an anisotropic form in $I_{q}^{n} F$ . For $n = 1$ it recaptures the definition of $u(F)…

Fixed Points of Polynomials over Division Rings

Adam Chapman, Solomon Vishkautsan

We study the discrete dynamics of standard (or left) polynomials $f (x)$ over division rings $D$ . We define their fixed points to be the points $I^» \in D$ for which $f^{\circ n}(Î»)=…

Quaternion quadratic equations in characteristic 2

Adam Chapman

In this paper we present a solution for any standard quaternion quadratic equation, i.e. an equation of the form $z^{2} + I^¼ z + I^½ = 0$ where $I^¼$ and $I^½$ belong to some quaternion divisi…

Central Products of Cayley-Dickson Loops

Adam Chapman, Ilan Levin

This paper studies the triviality of commutators in central products of Cayley-Dickson loops. Two immediate outcomes of this study are (1) the construction of a sequence of non-com…

Classes in $H_{p^{m}}^{n + 1} (F)$ of lower exponent

Adam Chapman, Daniel Krashen, Kelly McKinnie

Let $F$ be a field of characteristic $p > 0$ . We prove that if a symbol $A = I ¨ \otimes I^_{1}^{2} \otimes \dots \otimes I^_{n}^{2}$ in $H_{p^{m}}^{n + 1} (F)$ is of exponent dividing $p^{m - 1}$ , then it…

Symbol Length of $p$ -Algebras of Prime Exponent

Adam Chapman

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $char (F) = p$ is a finite integer $d$ gr…

$p$ -Central Subspaces of Central Simple Algebras

Adam Chapman

We study central simple algebras in various ways, focusing on the role of $p$ -central subspaces. The first part of my thesis is dedicated to the study of Clifford algebras. The sta…

Tensor Products of Cyclic Algebras of Degree 4 and their Kummer Subspaces

Adam Chapman, Charlotte Ure

We prove that the maximal dimension of a Kummer space in the generic tensor product of $n$ cyclic algebras of degree 4 is $4 n + 1$ .

math.CO2024

Loops with involution and the Cayley-Dickson doubling process

Adam Chapman, Ilan Levin, Uzi Vishne +1

We develop a theory of loops with involution. On this basis we define a Cayley-Dickson doubling on loops, and use it to investigate the lattice of varieties of loops with involutio…

math.AC2019

Linkage of Pfister forms over $C (x_{1}, \dots, x_{n})$

Adam Chapman, Jean-Pierre Tignol

In this note, we prove the existence of a set of $n$ -fold Pfister forms of cardinality $2^{n}$ over $C (x_{1}, \dots, x_{n})$ which do not share a common $(n - 1)$ -fold factor. This…

Totally ramified subfields of $p$ -Algebras

Adam Chapman, S. Srimathy

We conjecture that a $p$ -algebra over a complete discrete valued field $K$ contains a totally ramified purely inseparable subfield if and only if it contains a totally ramified cyc…

Fixed points and orbits in skew polynomial rings

Adam Chapman, Elad Paran

We study orbits and fixed points of polynomials in a general skew polynomial ring $D [x, I ¨ , I^´]$ . We extend results of the first author and Vishkautsan on polynomial dynamics in $D[…

math.RA2018

Triple Linkage of Quadratic Pfister Forms

Adam Chapman, Andrew Dolphin, David B. Leep

Given a field $F$ of characteristic 2, we prove that if every three quadratic $n$ -fold Pfister forms have a common quadratic $(n - 1)$ -fold Pfister factor then $I_{q}^{n + 1} F = 0$ . As a…

math.KT2026

Sums of two symbols in $K_{2} (F) /2 K_{2} (F)$ in characteristic two

Demba Barry, Adam Chapman, Ahmed Laghribi

In this paper, study sums $A = {a, b}_{2} + {c, d}_{2}$ of two symbols in $K_{2} (F) /2 K_{2} (F)$ when $char (F) = 2$ . We first prove a chain lemma that connects $A$ to $B=\{Î±,Î²\}…

math.RA2011

General Polynomials over Division Algebras and Left Eigenvalues

Adam Chapman

In this paper, we present an isomorphism between the ring of general polynomials over a division ring of degree $p$ over its center $F$ and the group ring of the free monoid with $…

cond-mat.mes-hall2019

Classification of Crystalline Topological Insulators and Superconductors with Point Group Symmetries

Eyal Cornfeld, Adam Chapman

Crystalline topological phases have recently attracted a lot of experimental and theoretical attention. Key advances include the complete elementary band representation analyses of…

Square-Central and Artin-Schreier Elements in Division Algebras

Demba Barry, Adam Chapman

We study the behavior of square-central elements and Artin-Schreier elements in division algebras of exponent 2 and degree a power of 2. We provide chain lemmas for such elements i…

math.RA2023

Chain Lemma, Quadratic Forms and Symbol Length

Adam Chapman, Ilan Levin

We want to bound the symbol length of classes in $_{2^{m - 1}} B r (F)$ which are represented by tensor products of 5 or 6 cyclic algebras of degree $2^{m}$ . The main ingredients are th…

Kummer Spaces in Cyclic Algebras of Prime Degree

Adam Chapman, David J. Grynkiewicz, Eliyahu Matzri +2

We classify the monomial Kummer subspaces of division cyclic algebras of prime degree $p$ , showing that every such space is standard, and in particular the dimension is no greater…

Symbol $p$ -Algebras of Prime Degree and their $p$ -Central Subspaces

Adam Chapman, Michael Chapman

We prove that the maximal dimension of a $p$ -central subspace of the generic symbol $p$ -algebra of prime degree $p$ is $p + 1$ . We do it by proving the following number theoretic fac…

Essential Dimension of Central Simple Algebras when the Characteristic is Bad

Adam Chapman, Kelly McKinnie

This is a survey of the existing literature, the state of the art, and a few minor new results and open questions regarding the essential dimension of central simple algebras and f…

math.RA2018

Kato-Milne Cohomology and Polynomial Forms

Adam Chapman, Kelly McKinnie

Given a prime number $p$ , a field $F$ with $char (F) = p$ and a positive integer $n$ , we study the class-preserving modifications of Kato-Milne classes of decomposable…

On the generalized Clifford algebra of a monic polynomial

Adam Chapman, Jung-Miao Kuo

In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial $Î¦(Z,X_1,\dots,X_n)=Z^d-\sum_{k…

math.AC2021

Linkage of sets of Quaternion Algebras in characteristic 2

Adam Chapman

This note contains two new observations on the linkage properties of quaternion algebras over fields of characteristic 2: first, that a 3-linked field need not be 4-linked (a case…

math.NT2025

Mixed multiquadratic splitting fields

Fatma Kader BingÃ¶l, Adam Chapman, Ahmed Laghribi

We study mixed multiquadratic field extensions as splitting fields for central simple algebras of exponent $2$ in characteristic $2$ . As an application, we provide examples of none…

math.AC2025

5-dimensional minimal quadratic and bilinear forms over function fields of conics

Adam Chapman, Ahmed Laghribi

Over a field of characteristic 2, we give a complete classification of quadratic and bilinear forms of dimension 5 that are minimal over the function field of an arbitrary conic. T…

Asymptotic Brauer $p$ -Dimension

Adam Chapman, Kelly McKinnie

We define and compute $ABrd_{p} (F)$ , the asymptotic Brauer $p$ -dimension of a field $F$ , in cases where $F$ is a rational function field or Laurent series field. $\ope…

math.RA2019

Polynomial Equations over Octonion Algebras

Adam Chapman

In this paper we present a complete method for finding the roots of all polynomials of the form $I ¨ (z) = c_{n} z^{n} + c_{n - 1} z^{n - 1} + \dots + c_{1} z + c_{0}$ over a given octonion division algeb…

math.RA2020

Field of Iterated Laurent Series and its Brauer Group

Adam Chapman

The symbol length of $_{p} B r (k ((I^\pm_{1})) \dots ((I^\pm_{n})))$ for an algebraically closed field $k$ of $char (k) \neq = p$ is known to be $⌊ \frac{n}{2} ⌋$ .…

Symbol Length of Classes in Milnor $K$ -groups

Adam Chapman

Given a field $F$ , a positive integer $m$ and an integer $n \geq 2$ , we prove that the symbol length of classes in Milnor's $K$ -groups $K_{n} F / 2^{m} K_{n} F$ that are equivalent to singl…

math.AC2018

Common Slots of Bilinear and Quadratic Pfister Forms

Adam Chapman

We show that over any field $F$ of $char (F) = 2$ and 2-rank $n$ , there exist $2^{n}$ bilinear $n$ -fold Pfister forms that have no slot in common. This answers a question…

Standard Polynomial Equations over Division Algebras

Adam Chapman, Casey Machen

Given a central division algebra $D$ of degree $d$ over a field $F$ , we associate to any standard polynomial $I ¨ (z) = z^{n} + c_{n - 1} z^{n - 1} + \dots + c_{0}$ over $D$ a "companion polynomial"…