Summary: REMARKS ON A PROBLEM OF EISENSTEIN
ROGER C. ALPERIN
Abstract. The fundamental unit of Z[

N] for square-free N = 5 mod 8
is either or 3
where denotes the fundamental unit of the maximal
order of Q(

N). We give infinitely many examples for each case.
1. Introduction
For N square-free, the ring of integers ON of a real quadratic field Q(

N)
has an infinite cyclic group of units of index 2. The generator for this
subgroup is the fundamental unit. The ring of integers ON has a subring
AN = Z[

N]; this is a proper subring if and only if N = 1 mod 4. The
subring also has an infinite cyclic subgroup of units generated by e; it is

Source: Alperin, Roger C. - Department of Mathematics, San Jose State University

Collections: Mathematics