Fundamental Discriminant - Connection With Quadratic Fields

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).