Ramification in relative quadratic extensions and fundamental units of real quadratic fields

Let F=QQ(d)be areal quadratic field ande=x+yd the fundamental unit of F satisfying NF/Q(e)=1.Some connections between the ramification properties for dyadic prime ideals in quadratic extension F(e)/Fand congruence properties of x,y were established.As a corollary,some congruence properties about x,y were given when d=p1…pror 2p1…pr with p1≡…≡pr≡1mod 4being distinct prime numbers.