Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1911.02368

The Brauer Group of Rational Numbers

Haiyu Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this project, we will study the Brauer group that was first defined by R. Brauer. The elements of the Brauer group are the equivalence classes of finite dimensional central simple algebra. Therefore understanding the structure of the Brauer group of a field is equivalent to a complete classification of finite dimensional central division algebras over the field. One of the important achievements of algebra and number theory in the last century is the determination of Br(Q), the Brauer group of rational numbers. The aim of this dissertation is to review this project, i.e., determining Br(Q). There are three main steps. The first step is to determine Br(R), the Brauer group of real numbers. The second step is to identify Br(k_\nu), the Brauer group of the local fields. The third step is to construct two maps Br(Q) to Br(R) and Br(Q) to Br(Q_p) and to use these two maps to understand Br(Q). This dissertation completed the first two steps of this enormous project. To the author's knowledge, in literature there is no document including all the details of determining Br(Q) and most of them are written from a advanced perspective that requires the knowledge of class field theory and cohomology. The goal of this document is to develop the result in a relatively elementary way. The project mainly follows the logic of the book [6], but significant amount of details are added and some proofs are originated by the author, for example, 1.2.6, 1.4.2(ii), 4.2.6, and maximality and uniqueness of 5.5.12.

Explore related subjects

math.RAmath.GRmath.HOmath.NT

Keep this discovery

BibTeXRIS

Haiyu Chen. 2019-11-02. The Brauer Group of Rational Numbers. https://arxiv.org/abs/1911.02368

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation

We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.

math.RA↗
arXiv2026

Quasilinear multiplication in the real Cayley--Dickson tower

Direct evaluation of the defining product in the real Cayley--Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge16$ and than the uniform Cariow--Cariowa method for $N\ge32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. In single-core benchmarks against direct multiplication and the uniform Cariow--Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.

math.RA↗
arXiv2026

Graded classification of Leavitt path algebras in terms of strong shift equivalence

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

math.RA↗