An introduction to noncommutative arithmetic

This is a colloquium style talk. We begin by introducing local and global fields, then look at the theory of central simple algebras over these fields. Our main focus is on maximal orders in the aforementioned algebras. In the words of the late Irving Reiner, "The theory of maximal orders is of interest in its own right, and is essentially the study of "noncommutative arithmetic". The beauty of the subject stems from the fascinating interplay between the arithmetical properties of orders, and the algebraic properties of the algebras containing them." Finally, we introduce the ideal class group of a maximal order and show how maximal left ideals are distributed in classes. This last part is a refinement of a special case of Collin J. Bushnell and Irving Reiner's prime ideal theorem for noncommutative arithmetic (1982).