On þ-adic Expansions of Algebraic Integers
Hsing-Hau Chen, Ming-Deh A. Huang · 2015
It is well known that every rational integer has a finite or periodic p-adic expansion. In this paper a more general notion of Þ-adic expansion is introduced for algebraic integers, where given a number field K and a principal prime ideal Þ in K, a different choice of generator for Þ is allowed in each stage of the expansion. With the notion of Þ-adic expansion, we prove that there is always a finite or periodic Þ-adic expansion for every algebraic integer. Moreover, we prove a bound on the periodicity of the Þ-adic expansion that depends only on the number field K and the prime ideal Þ. The proof yields an algorithm for constructing such a Þ-adic expansion for elements in the ring O of algebraic integers of K, through finding an approximation to the closest vector on the lattice spanned by the unit group of O.