Mathematics — Number Theory · class-number-one-negative-even-discriminants
Problem
In this extended exercise the goal will be to show that the only negative even discriminants with class number 1 are −4,−8,−12,−16, and −28. (Of these only −4 and −8 are fundamental discriminants.) The strategy will be to exhibit an explicit reduced primitive form Q different from the principal form x2+dy2 for each discriminant −4d with d>4 except d=7. This will be done by breaking the problem into several cases, where in each case a form Q will be given and you are to show that this form has the desired properties, namely it is of discriminant −4d, primitive, reduced, and different from the principal form. You should also check that the cases considered cover all possibilities.
(a) Suppose d is not a prime power. Then it can be factored as d=ac where 1<a<c and a and c are coprime. In this case let Q be the form ax2+cy2.
(b) The form ax2+2xy+cy2 will work provided that d+1 factors as d+1=ac where a and c are coprime and 1<a<c. If d is odd, for example a power of an odd prime, then d+1 is even so it has such a factorization d+1=ac unless d+1=2n.
(c) If d=2n the cases we need to consider are n≥3 since d>4. When n=3 take Q to be 3x2+2xy+3y2 and when n≥4 take Q to be 4x2+4xy+(2n−2+1)y2.
(d) When d+1=2n the cases of interest are n≥3. When n=3 we have d=7 which is one of the allowed exceptions with class number 1. When n=4 we have d=15 and 3x2+5y2 works as in part (a). When n=5 we have d=31 and we take the form 5x2+4xy+7y2. When n≥6 we use the form 8x2+6xy+(2n−3+1)y2.