c © 2004 Kluwer Academic Publishers. Manufactured in The Netherlands. 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 + √D)/2: x, y ∈ 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. Key words: quadratic units of norm −1, quadratic elements of norm ±2 2000 AMS Mathematics Subject Classification: Primary—11R11, 11R27 1