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Fundamental Theorem of Genera

Consider $h_+(d)$ proper equivalence classes of forms with discriminant $d$ equal to the field discriminant, then they can be subdivided equally into $2^{r-1}$ genera of $h_+(d)/2^{r-1}$ forms which form a subgroup of the proper equivalence class group under composition (Cohn 1980, p. 224), where $r$ is the number of distinct prime divisors of $d$. This theorem was proved by Gauß in 1801.


Arno, S.; Robinson, M. L.; and Wheeler, F. S. ``Imaginary Quadratic Fields with Small Odd Class Number.''

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

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

© 1996-9 Eric W. Weisstein