Arithmetic matrices for number fields I

Abstract: 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 at least a significant proportion 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 several algebraic integers at once might be more efficiently implemented using methods of efficient matrix multiplication. If we are multiplying an algebraic integer in a field of degree $n$ by each of $n$ algebraic integers, then this can be done in $O \left( n^{\log_2(7)} \right)$ multiplications. The matrix identities given here generalize Brahmagupta's identity upon taking determinants, which are multiplicative.