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 ε 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 ε ≡ ±1 (mod π2). In this paper we determine ε modulo π3 and modulo λ3 using only elementary techniques. This determination extends a recent result of Mastropietro, which was proved using class field theory.

Additional Metadata
Keywords Quadratic elements of norm ±2, Quadratic units of norm -1
Persistent URL dx.doi.org/10.1023/B:RAMA.0000012427.39580.ab
Journal Ramanujan Journal
Evans, R.J. (Ronald J.), Kaplan, P. (Pierre), & Williams, K.S. (2003). Congruences for quadratic units of norm -1. Ramanujan Journal, 7(4), 449–453. doi:10.1023/B:RAMA.0000012427.39580.ab