Is there any way to determine whether the fundamental unit of a quadratic field has negative or positive norm, except by actually computing the unit to all of its (many) digits? And, similarly, when $d$ is 1 mod 4, the ring of integers will have a basis with denominators, but sometimes the unit does belong to ${\mathbb Z} [\sqrt d]$--is there any way to tell "in advance" if that will happen, i.e., any way other than computing the unit explicitly? (I can't find any such methods in books, but of course that doesn't prove they don't exist.)

Is there any way to determine whether the fundamental unit of a quadratic field has negative or positive norm, except by actually computing the unit to all of its (many) digits? And, similarly, when $d$ is 1 mod 4, the ring of integers will have a basis with denominators, but sometimes the unit does belong to ${\mathbb Z} [\sqrt d]$—is there any way to tell "in advance" if that will happen, i.e., any way other than computing the unit explicitly? (I can't find any such methods in books, but of course that doesn't prove they don't exist.)