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.

# Fundamental Theorem of Genera

## See also

Form Genus, Genus Theorem## Explore with Wolfram|Alpha

## 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.

## Referenced on Wolfram|Alpha

Fundamental Theorem of Genera## Cite this as:

Weisstein, Eric W. "Fundamental Theorem of
Genera." From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalTheoremofGenera.html