Related papers: Symbol length in positive characteristic

We bound the symbol length of elements in the Brauer group of a field $K$ containing a $C_m$ field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a $C_m$…

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 provide an explicit computation of the resulting cyclic algebra…

Let $p$ be a prime integer, $1\leq s\leq r$ integers, $F$ a field of characteristic $p$. Let $\cat{Dec}_{p^r}$ denote the class of the tensor product of $r$ $p$-symbols and $\cat{Alg}_{p^r,p^s}$ denote the class of central simple algebras…

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $\operatorname{char}(F)=p$ is a finite integer $d$ greater than 1 then the symbol length of $p$-algebras of…

In this paper we study division algebras over the function fields of curves over $\Q_p$. The first and main tool is to view these fields as function fields over nonsingular $S$ which are projective of relative dimension 1 over the $p$ adic…

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…

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

We show that every central simple algebra A over a field k is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field (that is- A is Hopf Schur). If the characteristic of the field is zero, or if the algebra…

Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,\mu_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the…

In characteristic two, it is shown that a central simple algebra of degree equal to a power of two with anisotropic orthogonal involution is totally decomposable, if it becomes either anisotropic or metabolic over all extensions of the…

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^m$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell+1$ when $k$ is perfect. We do this by observing…

Let A be a central simple algebra over a field F. Let k_1,\ldots, k_r be cyclic extensions of F such that k_1\otimes_F\cdots \otimes_F k_r is a field. We investigate conditions under which A is a tensor product of symbol algebras where each…

The symbol length of ${_pBr}(k(\!(\alpha_1)\!)\dots(\!(\alpha_n)\!))$ for an algebraically closed field $k$ of $\operatorname{char}(k) \neq p$ is known to be $\lfloor \frac{n}{2} \rfloor$. We prove that the symbol length for the case of…

For each n>0, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the…

For a $p$-permutation equivalence between two block algebras of finite groups, we introduce new square diagrams that link the $p$-permutation equivalence via the Brauer construction to local equivalences between stabilizers of corresponding…

We study the common splitting fields of symbol algebras of degree $p^m$ over fields $F$ of $\operatorname{char}(F)=p$. We first show that if any finite number of such algebras share a degree $p^m$ simple purely inseparable splitting field,…

Let $K$ be a complete discretely valued field of rank one, with residue field $\Q_p$. It is well known that period equals index in $\Br(K)$. We prove that when $p=2$ there exist noncyclic $K$-division algebras of every $2$-power degree…

The Artin-Schreier polynomial $Z^p - Z - a$ is very well known. Polynomials of this type describe all degree $p$ (cyclic) Galois extensions over any commutative ring of characteristic $p$. Equally attractive is the associated Galois action.…

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 fact: let $\{s_1,\dots,s_{p+1}\}$ be $p+1$ distinct nonzero…

We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets…