Fundamental discriminant

In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax² + bxy + cy² is a quadratic form with integer coefficients, then D = b² − 4ac is the discriminant of Q(x, y). Conversely, every integer D with D ≡ 0, 1 (mod 4) is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as discriminants in this theory. Every discriminant may be written as

D = D₀f²

with D₀ a discriminant and f a positive integer. A discriminant D is called a fundamental discriminant if f = 1 in every such decomposition. Conversely, every discriminant D ≠ 0 can be written uniquely as D₀f² where D₀ is a fundamental discriminant. Thus, fundamental discriminants play a similar role for discriminants as prime numbers do for all integers.

There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, D is a fundamental discriminant if, and only if, one of the following statements holds

D ≡ 1 (mod 4) and is square-free,
D = 4m, where m ≡ 2 or 3 (mod 4) and m is square-free.

The first ten positive fundamental discriminants are:

1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 (sequence A003658 in OEIS).

The first ten negative fundamental discriminants are:

−3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence A003657 in OEIS).

Connection with quadratic fields

There is a connection between the theory of integral binary quadratic forms and the arithmetic of quadratic number fields. A basic property of this connection is that D₀ is a fundamental discriminant if, and only if, D₀ = 1 or D₀ is the discriminant of a quadratic number field. There is exactly one quadratic field for every fundamental discriminant D₀ ≠ 1, up to isomorphism.

Caution: This is the reason why some authors consider 1 not to be a fundamental discriminant. One may interpret D₀ = 1 as the degenerated "quadratic" field Q (the rational numbers).

Factorization

Fundamental discriminants may also be characterized by their factorization into positive and negative prime powers. Define the set

$S=\{-8, -4, 8, -3, 5, -7, -11, 13, 17, -19,\; \ldots\}$

where the prime numbers ≡ 1 (mod 4) are positive and those ≡ 3 (mod 4) are negative. Then, a number D₀ ≠ 1 is a fundamental discriminant if, and only if, it is the product of pairwise relatively prime members of S.

References

Henri Cohen (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics. 138. Berlin, New York: Springer-Verlag. ISBN 3-540-55640-0. MR 1228206.
Duncan Buell (1989). Binary quadratic forms: classical theory and modern computations. Springer-Verlag. p. 69. ISBN 0-387-97037-1.
Don Zagier (1981). Zetafunktionen und quadratische Körper. Berlin, New York: Springer-Verlag. ISBN 978-3540106036.