Fundamental discriminant

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In mathematics, a fundamental discriminant d is an integer that is the discriminant of a quadratic field. There is exactly one quadratic field with given discriminant, up to isomorphism.

There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, d is a fundamental discriminant if, and only if, d ≠ 1 and either

d\equiv 1 \mbox{ (mod }4)\mbox{ and is square-free}

or

d=4m,\mbox{ where }m\equiv 2\mbox{ or }3 \mbox{ (mod }4)\mbox{ and is square-free}.

The first ten positive fundamental discriminants are:

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)

[edit] 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 is a fundamental discriminant if and only if it is the product of pairwise relatively-prime members of S.

[edit] References

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