Quadratic Fields
Computing the Structure of Residue Rings
The structure of the unit group of a residue ring of the maximal order, and its computation by decomposition and lifting.
Engineering / MathematicsQuadratic Fields2 min readKV-MATH-0609
The unit group of the maximal order modulo an ideal generalises the integers modulo n. Its structure is needed for ray class groups, for conductor computations and for constructing characters.
Decomposition
By the Chinese remainder theorem the residue ring decomposes over the prime power components of the modulus, and the unit group decomposes correspondingly.
The prime case
Modulo a prime ideal, the residue ring is a finite field of size the norm, so the unit group is cyclic of order one less than the norm.
Prime powers
Modulo a higher power, the unit group has a cyclic part coming from the residue field and a filtration part coming from the successive quotients of powers of the prime.
| Component | Structure |
|---|---|
| Residue field part | Cyclic of order the norm minus one |
| Higher terms | An abelian group of order a power of the underlying rational prime |
| Wild case | When the residue characteristic divides the ramification index, the filtration structure is more intricate |
Computation
Computing the residue ring unit group
- Factor the modulusInto prime powers — see valuations.
- Handle each primeCompute the residue field part and the filtration part separately.
- Find generatorsLift generators of the residue field part; take one plus a uniformiser power for the filtration part.
- RecombineAssemble by the Chinese remainder theorem.
- NormaliseReduce to invariant factors by Smith normal form.
Archimedean places
Applications
Ray class groups generalise the ideal class group by imposing congruence conditions, and are the objects classified by class field theory. The computation above is the first step in constructing them, and the same machinery gives conductors and character groups.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.2. 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.