The minimal Euclidean function on the Gaussian integers

Abstract

In 1949, Motzkin proved that every Euclidean domain R has a minimal Euclidean function, ϕR. He showed that when R=Z, the minimal Euclidean function is ϕZ(x)=⌊log2|x|⌋. For over seventy years, ϕZ has been the only example of an explicitly-computed minimal Euclidean function for the ring of integers of a number field. We give the first explicitly-computed minimal Euclidean function in a non-trivial number field, ϕZ[i], which also computes the length of the shortest possible (1+i)-ary expansion of any Gaussian integer. We then present an algorithm that uses ϕZ[i] to compute minimal (1+i)-ary expansions of Gaussian integers. We solve these problems using only elementary methods.