Congruences for Quadratic Units of Norm −1

Abstract

Let D ≡ 1 (mod 4) be a positive integer. Let R be the ring {x + y(1 + $\sqrt D$)/2 : x, y ∈ $\mathbb{Z}$}. Suppose that R contains a unit ∈ of norm −1 as well as an element π of norm 2, and thus an element λ of norm −2. It is not hard to see that ∈ ≡ ±1(mod π²). In this paper we determine ∈ modulo π³ and modulo λ³ using only elementary techniques. This determination extends a recent result of Mastropietro, which was proved using class field theory.