Arithmetic matrices for number fields I

Samuel A. Hambleton

Published 2018-08-29Version 1

We provide a simple way to add, multiply, invert, and take traces and norms of algebraic integers of a number field using integral matrices. With formulas for the integral bases of the ring of integers of numbers fields, we obtain explicit formulas for these matrices and discuss their generalization. These results are useful in proving statements about the particular number fields they work in. We give a meaningful diagonalization that is helpful in this regard and can be used to define such matrices in general. The matrix identities provided suggest that consideration of numerical methods in linear algebra may have applications in algebraic number theory. Multiplication of algebraic integers might be more efficiently implemented using methods of efficient matrix multiplication. The matrix identities given here generalize Brahmagupta's identity upon taking determinants, which are multiplicative.