Number Fields and Algebraic Numbers
Discriminants and Integral Bases
The discriminant of a basis, the field discriminant, and the index-squared relation that governs maximal order computation.
Engineering / MathematicsNumber Fields and Algebraic Numbers2 min readKV-MATH-0579
The discriminant is the central invariant of a number field. It measures ramification, bounds the class number, and — through the index-squared relation — determines what work is required to find the maximal order.
Discriminant of a basis
Equivalently, it is the square of the determinant of the matrix of conjugates. Both formulations are computable; the trace form is exact and preferred.
The index-squared relation
Changing basis by an integer matrix multiplies the discriminant by the square of the determinant. For an order inside the maximal order, that determinant is the index.
The procedure
Computing the field discriminant and an integral basis
- Compute the polynomial discriminantBy resultant with the derivative.
- Factor itOr at least find its square factors. This is often the expensive step.
- Test each square factorFor each prime whose square divides the discriminant, test whether the order is maximal at that prime — see the Dedekind criterion.
- EnlargeWhere it is not, enlarge the order — see Round 2.
- Divide outThe field discriminant is the polynomial discriminant divided by the index squared.
The factoring obstacle
Partial results
If the discriminant cannot be factored completely, an order maximal at all known primes can still be computed, with the unfactored part flagged. Results depending only on those primes remain valid; results requiring full maximality do not.
Ramification
A prime ramifies in the field exactly when it divides the field discriminant. The discriminant therefore encodes the complete ramification data — see decomposition and ramification.
Bounds
The discriminant bounds the class number and the regulator through the Minkowski bound and the analytic class number formula, which is why it appears in the complexity of class group computation. See Minkowski and Bach bounds.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.4.1. 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.