p-integral bases of algebraic number fields

Abstract

Let K be an algebraic number field of degree n, let p be a rational prime, and let α ∈ K. If ν_P (α) ≥ 0 for each prime ideal P of K such that P\pO_K then α is called a p-integral element of K. Let {w₁, w₂,..., w_n} be a basis of K over Q, where each w_i (1 ≤ i ≤ n) is a p-integral element of K. If every p-integral element α of K is given as α = a₁w₁ + a₂w₂ + ⋯ + a_nw_n, where the a_i are p-integral elements of Q, then {w₁, w₂,..., w_n} is called a p-integral basis of K. The properties of p-integral bases of an algebraic number field K are developed, and used to show how an integral basis of K can be obtained from its p-integral bases.