Orders, Ideals and Prime Decomposition
Module Representation by Hermite Normal Form
Representing ideals and modules as Hermite normal form matrices relative to an integral basis, with a common denominator.
Engineering / MathematicsOrders, Ideals and Prime Decomposition2 min readKV-MATH-0585
The standard representation of an ideal is a matrix in Hermite normal form giving a basis relative to the integral basis of the order, together with a denominator. It is canonical, which is what makes equality testing possible.
The representation
Invariants maintained
What can be read directly
| Quantity | From the representation |
|---|---|
| Norm | Product of the diagonal entries, divided by the denominator to the field degree |
| Integrality | Denominator equal to one |
| Membership | Reduce the element's coordinate vector against the triangular basis |
| Containment | Test each basis vector of one for membership in the other |
Stability under the order
Cost
Alternatives
Two-element representation
Compact and fast for multiplication, but not canonical. See two-element representation.
Factored representation
Store the prime factorisation with exponents. Ideal for multiplication and norms, poor for addition, and requires the factorisation to be known.
Reduced representative
For class group work, a small representative in the ideal class is often more useful than the ideal itself — see ideal reduction.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.7.1. 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.