 
Summary: REMARKS ON A PROBLEM OF EISENSTEIN
ROGER C. ALPERIN
Abstract. The fundamental unit of Z[
N] for squarefree 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 squarefree, 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
