Quadratic Fields
Binary Quadratic Forms and the Ideal Correspondence
The dictionary between binary quadratic forms and ideals of a quadratic order, and why the form language is computationally preferable.
Engineering / MathematicsQuadratic Fields2 min readKV-MATH-0601
Ideals of a quadratic order correspond to binary quadratic forms of the same discriminant. The two languages describe the same objects, and the form language is more compact and computationally faster.
The dictionary
| Ideal language | Form language |
|---|---|
| Ideal of the order | Binary quadratic form |
| Norm of the ideal | First coefficient of the form |
| Ideal multiplication | Composition of forms |
| Ideal class | Equivalence class of forms |
| Class group | Form class group |
| Reduction of an ideal | Reduction of a form |
| Principal ideal | Principal form |
Definite and indefinite
- Positive definite
- Negative discriminant. Corresponds to an imaginary quadratic field. Each class contains exactly one reduced form.
- Indefinite
- Positive discriminant. Corresponds to a real quadratic field. Each class contains a cycle of reduced forms.
Equivalence
Two forms are equivalent when related by a change of variables with determinant one. Equivalence classes correspond to ideal classes, and the number of classes is the class number.
Non-fundamental discriminants
Operations
Reduction is covered in positive definite reduction and indefinite reduction; multiplication in composition.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 5.2. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.