Orders, Ideals and Prime Decomposition
Prime Decomposition when p Does Not Divide the Index
Decomposing a prime that does not divide the index, by factoring the defining polynomial modulo that prime.
Engineering / MathematicsOrders, Ideals and Prime Decomposition2 min readKV-MATH-0590
When a prime does not divide the index of the equation order in the maximal order, its decomposition is read directly off the factorisation of the defining polynomial modulo that prime. This covers almost every prime.
The theorem
Factor the defining polynomial modulo p into irreducible factors with multiplicities. Each factor corresponds to a prime ideal above p, with residue degree the factor's degree and ramification index its multiplicity.
Prime decomposition in the simple case
- Verify the conditionConfirm p does not divide the index — equivalently, that p squared does not divide the polynomial discriminant, or use the Dedekind criterion.
- Factor modulo pUsing the finite field pipeline — see factorisation strategy.
- Build the idealsEach prime is generated by p and the lift of the corresponding factor evaluated at the generator.
- Record e and fMultiplicity and degree of each factor.
- VerifyConfirm the sum of e times f equals the field degree.
Checking the condition
| Test | Cost | Conclusiveness |
|---|---|---|
| p squared does not divide the polynomial discriminant | Requires the discriminant only | Sufficient, not necessary |
| p does not divide the index | Requires the maximal order | Exact |
| Dedekind criterion | One polynomial GCD computation modulo p | Exact, and does not need the maximal order |
Ramification detection
A repeated factor modulo p indicates ramification. Since repeated factors are found by squarefree factorisation, the ramification structure emerges from the first stage of the factorisation pipeline at no extra cost.
When it fails
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.8.2. 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.