Os Quatérnios

Abstract

In this work we describe the construction of the quaternions algebra H, from two different perspectives. The first follows the ideas of Hamilton himself, who introduced quaternions in 1843, and the second, as a subalgebra of certain complex matrices. We study the fundamental properties of H, both algebraic and geometric. In particular we prove that H is an associative division algebra of dimension 4 over R. Our study culminates by presenting a detailed proof of the famous Frobenius’s Theorem, which characterizes finite dimensional associative division algebras over R. According to said theorem, each such algebra is isomorphic to one of the following algebras: (i) R, the algebra of real numbers; (ii) C, the algebra of complex numbers; (iii) H, the algebra of quaternions.