Fundamental Theorem of Genera

Consider  proper equivalence classes of forms with discriminant  equal to the field discriminant, then they can be subdivided equally into  genera of  forms which form a subgroup of the proper equivalence class group under composition (Cohn 1980, p. 224), where  is the number of distinct prime divisors of . This theorem was proved by Gauss in 1801.

REFERENCES:

Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." http://www.math.uiuc.edu/Algebraic-Number-Theory/0009/.

Cohn, H. Advanced Number Theory. New York: Dover, 1980.

Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT: Yale University Press, 1966.