Composition algebras over a ring of fractions
Let \(k\) be a field of characteristic distinct from 2, and let \(R\) be the ring of fractions defined by \[ R=\left\{{g(t) \over f(t)^j} \in k(t) \mid j\geq 0,\;g(t)\in k[t] \text{ of }\deg \leq 2j \right\} \simeq k[x_0,x_1]_{(x^2_0 -ax_1^2)} \] with \(f(t)=t^2- a\in k[t]\) and \(a\in k^\times\). The author considers composition algebras without zero divisors over the ring \(R\), lists and partly classifies them. She gives also some examples for special base fields \(k\). Her main results are: The Picard group of a nonsplit torus defined over \(k\) contains at most two elements, one of them the torus itself considered as an \(R\)-module. Every nonsplit quaternion algebra over \(R\) not defined over \(k\) can be realized as a nonclassical Cayley-Dickson doubling of a torus defined over \(k\) or, equivalently, as a classical Cayley-Dickson doubling of a torus not defined over \(k\). Octonion algebras not defined over \(k\) can either be realized as a nonclassical Cayley-Dickson doubling of some quaternion algebra, or, equivalently, as a classical Cayley-Dickson doubling of a quaternion algebra not defined over \(k\).
composition algebras without zero divisors
scientific article; zbMATH DE number 1014020
scientific article