Fundamental discriminant

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:$dequiv\; 1\; mbox\{\; (mod\; \}4)mbox\{\; and\; is\; square-free\}$or:$d=4m,mbox\{\; where\; \}mequiv\; 2mbox\{\; or\; \}3\; mbox\{\; (mod\; \}4)mbox\{\; and\; is\; square-free\}.$

The first ten positive fundamental discriminants are:: 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 OEIS|id=A003658

The first ten negative fundamental discriminants are:: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 OEIS|id=A003657

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".

References

*cite book | author=Henri Cohen | authorlink=Henri Cohen | title=A Course in Computational Algebraic Number Theory | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | volume=138 | isbn=3-540-55640-0 | id=MathSciNet | id = 1228206 | year=1993
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