In this paper, we present a new method for solving standard quaternion equations. Using this meth... more 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 equation, and provide an explicit solution for the cubic quaternion equation, as long as the equation has at least one pure imaginary root. We also discuss the number of essential pure imaginary roots of a twosided quaternion polynomial.
Bulletin of the Belgian Mathematical Society - Simon Stevin, 2015
We show that if two division p-algebras of prime degree share an inseparable field extension of t... more 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 is in general not true. We also point out that sharing all the inseparable field extensions of the center does not imply sharing all the cyclic separable ones.
Proceedings of the American Mathematical Society, 2018
Given a field F of char(F) = 2, we define u n (F) to be the maximal dimension of an anisotropic f... more Given a field F of char(F) = 2, we define u n (F) to be the maximal dimension of an anisotropic form in I n q F. For n = 1 it recaptures the definition of u(F). We study the relations between this value and the symbol length of H n 2 (F), denoted by sl n 2 (F). We show for any ). As a result, if u(F) is finite then sl n 2 (F) is finite for any n, a fact which was previously proven when char(F) 2 by Saltman and Krashen. We also show that if sl n 2 (F) = 1 then u n (F) is either 2 n or 2 n+1 .
Proceedings of the American Mathematical Society, 2017
We discuss the Kummer subspaces of tensor products of cyclic algebras, focusing mainly on the cas... more 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 general case, classify all the monomial spaces in the case of tensor products of cyclic algebras of degree 3 using graph theory, and provide an upper bound for the dimension in the generic tensor product of cyclic algebras of degree 3.
We prove that the maximal dimension of a Kummer space in the generic tensor product of n cyclic a... more We prove that the maximal dimension of a Kummer space in the generic tensor product of n cyclic algebras of degree 4 is 4n + 1.
We study central simple algebras in various ways, focusing on the role of $p$-central subspaces. ... more 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 standard Clifford algebra of a given form is the generic associative algebra containing a $p$-central subspace whose exponentiation form is equal to the given form. There is an old question as for whether these algebras have representations of finite rank over the center, and jointly with Daniel Krashen and Max Lieblich we managed to provide a positive answer. Different generalizations of the structure of the Clifford algebra are presented and studied in that part too. The second part is dedicated to the study of $p$-central subspaces of given central simple algebras, mainly tensor products of cyclic algebras of degree $p$. Among the results, we prove that $5$ is the upper bound for the dimension of 4-central subspaces of cyclic algebras of degree 4 containing pairs of standard generators. The third part is d...
We study the subfields of quaternion algebras that are quadratic extensions of their center in ch... more We study the subfields of quaternion algebras that are quadratic extensions of their center in characteristic $2$. We provide examples of the following: Two nonisomorphic quaternion algebras that share all their quadratic subfields, two quaternion algebras that share all the inseparable but not all the separable quadratic subfields, and two algebras that share all the separable but not all the inseparable quadratic subfields. We also discuss quaternion algebras over global fields and fields of Laurent series over a perfect field of characteristic $2$ and show that the quaternion algebras over these fields are determined by their separable quadratic subfields.
In this paper, we prove that for a given biquaternion algebra over a field of characteristic two,... more 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 that in each step at least one entry remains unchanged. If one requires that in each step two entries remain the same then their number increases to fifteen. We provide even more basic steps that in order to move from one symbol presentation to another one needs to use up to forty-five of them.
In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with res... more In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial Φ(Z, X 1 , . . . , of degree d in n + 1 variables over some field F . We completely determine its structure in the following cases: n = 2 and d = 3 and either char (F ) = 3, 2 for some r, t, e ∈ F . Except for a few exceptions, this algebra is an Azumaya algebra of rank nine whose center is the coordinate ring of an affine elliptic curve. We also discuss representations of arbitrary generalized Clifford algebras assuming the base field F is algebraically closed of characteristic zero.
We study the behavior of square-central elements and Artin-Schreier elements in division algebras... more 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 in division algebras over 2-fields F of cohomological 2dimension cd 2 (F) ≤ 2, and deduce a common slot lemma for tensor products of quaternion algebras over such fields. We also extend to characteristic 2 a theorem proven by Merkurjev for characteristic not 2 on the decomposition of any central simple algebra of exponent 2 and degree a power of 2 over a field F with cd 2 (F) ≤ 2 as a tensor product of quaternion algebras.
Abstract: Here we present a reduction of any quaternion standard polynomial equation into an equa... more Abstract: Here we present a reduction of any quaternion standard polynomial equation into an equation with two central variables and quaternion coefficients. If only pure imaginary roots are in demand, then the equation is with one central variable. As a result of this ...
In this paper, we present a solution for any standard quaternion quadratic equation, i.e. an equa... more In this paper, we present a solution for any standard quaternion quadratic equation, i.e. an equation of the form z2 + μz + ν = 0 where μ and ν belong to some quaternion division algebra Q over some field F, assuming the characteristic of F is 2.
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Papers by adam chapman