Quadratic Fields
Quadratic Field Discriminants and Integral Bases
Discriminants and integral bases of quadratic fields, given by closed formulas with no computation required.
Engineering / MathematicsQuadratic Fields2 min readKV-MATH-0599
Quadratic fields are the one family where every structural question has a closed-form answer. No maximal order algorithm is needed; the integral basis is given by a case split on a congruence.
Presentation
Every quadratic field is generated by the square root of a squarefree integer, positive for a real field and negative for an imaginary one.
Discriminant and integral basis
| Condition on m | Discriminant | Integral basis |
|---|---|---|
| m congruent to 1 modulo 4 | m | 1 and (1 + sqrt(m))/2 |
| Otherwise | 4m | 1 and sqrt(m) |
Fundamental discriminants
- Fundamental discriminant
- An integer that is the discriminant of some quadratic field: either one modulo four and squarefree, or four times a squarefree number that is two or three modulo four.
- Non-fundamental discriminant
- The discriminant of a non-maximal order. Valid for form theory but not a field discriminant.
- Conductor
- The factor relating a general discriminant to the fundamental one beneath it.
Signature and unit rank
| Field | Signature | Unit rank | Roots of unity |
|---|---|---|---|
| Imaginary quadratic | r1 = 0, r2 = 1 | 0 | Usually plus and minus one; more for the discriminants minus three and minus four |
| Real quadratic | r1 = 2, r2 = 0 | 1 | Plus and minus one only |
Why they are the right place to start
Every general phenomenon — class groups, units, regulators, reduction theory, sub-exponential methods — appears here in a setting concrete enough to compute by hand. See the quadratic pathway.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 5.1. 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.