Oswald N., Steuding J. Hurwitz's Lectures on the Number Theory of Quaternions

EMS Press, 2023. — 311 p.Quaternions are non-commutative generalizations of the complex numbers. They were discovered (or invented) by William Rowan Hamilton about 175 years ago. Their number-theoretical aspects were frst investigated by Rudolf Lipschitz in the 1880s. His approach, however, was revised and streamlined by the work of Adolf Hurwitz in 1896. The event of these two studies on quaternions appears to be typical for the state of the art of number theory at the turn of the twentieth century, where new concepts (e.g., ideals) were introduced for a better understanding of a generalized arithmetic of algebraic integers. The quaternion counterpart of integers is interesting not only for the parallels and differences to the ordinary integers and their counterparts in algebraic number felds, but their properties also shed light on the arithmetic of integers.