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Reduced Binary Quadratic Form


The binary quadratic form F=<a,b,c> is said to be reduced if the following conditions hold. Let D=b^2-4ac be the discriminant, then

1. If D is negative, F is reduced if |b|<=a<=c and if b>=0 whenever a=|b| or a=c, and F is called real.

2. If D is positive, F is reduced if |sqrt(D)-2|c||<b<sqrt(D), and F is called imaginary or positive definite.

Every imaginary binary quadratic form is equivalent to a unique reduced form and every real binary quadratic form is equivalent to a finite number of reduced forms.


See also

Binary Quadratic Form, Discriminant, Positive Definite Quadratic Form

This entry contributed by David Terr

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References

Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, pp. 226 and 257, 1993.

Referenced on Wolfram|Alpha

Reduced Binary Quadratic Form

Cite this as:

Terr, David. "Reduced Binary Quadratic Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ReducedBinaryQuadraticForm.html

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