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