Orders, Ideals and Prime Decomposition
Orders in Number Fields
Orders as subrings that are full-rank lattices, the equation order, the maximal order, and the index that separates them.
Engineering / MathematicsOrders, Ideals and Prime Decomposition2 min readKV-MATH-0583
An order is a subring of a number field that is also a lattice of full rank. The maximal one is the ring of integers, and it is the only order in which ideal factorisation behaves properly.
Definition
- Order
- A subring containing one, finitely generated as a module, spanning the field over the rationals.
- Equation order
- The order generated by the powers of the defining polynomial's root. Immediate from the presentation and generally not maximal.
- Maximal order
- The ring of all algebraic integers in the field. Unique, and contains every other order.
- Conductor
- The largest ideal of the maximal order contained in a given order. Measures how far the order is from maximal.
Why maximality matters
| Property | Maximal order | General order |
|---|---|---|
| Unique ideal factorisation | Holds | Fails |
| Every non-zero ideal invertible | Yes | Only those coprime to the conductor |
| Integrally closed | Yes | No |
| Class group | The ideal class group | A ring class group; coarser |
The index
An order sits inside the maximal order with finite index, and that index relates the two discriminants.
Representation
An order is represented by a basis expressed relative to the power basis of the defining polynomial, in Hermite normal form with a common denominator. Multiplication tables are precomputed from the multiplication matrices of the basis elements.
Computing the maximal order
The computation is the subject of the maximal order stream, via the Dedekind criterion and the Round 2 algorithm.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.6.1. 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.