Orders, Ideals and Prime Decomposition
Prime Decomposition: Theory and Ramification
How rational primes factor in the maximal order, ramification indices and residue degrees, and the degree relation that constrains them.
Engineering / MathematicsOrders, Ideals and Prime Decomposition2 min readKV-MATH-0589
A rational prime generates an ideal of the maximal order that factors into prime ideals. How it factors is the central local question about a number field, and it is completely determined by two integers per prime above.
The factorisation
- Ramification index e
- The exponent of the prime in the factorisation.
- Residue degree f
- The degree of the residue field of the prime over the field with p elements.
- Decomposition number g
- The number of distinct primes above p.
The named cases
| Case | Condition | Description |
|---|---|---|
| Split completely | g = n, all e and f equal one | n distinct primes, each of norm p |
| Inert | g = 1, e = 1, f = n | p remains prime in the order |
| Totally ramified | g = 1, e = n, f = 1 | A single prime appearing to the n-th power |
| Ramified | Some e greater than one | p divides the discriminant |
| Unramified | All e equal one | p does not divide the discriminant |
Ramification and the discriminant
The computational split
- Does p divide the index of the equation order?
- No — factor the defining polynomial modulo pSee the simple case
- Yes — the simple method fails
- Use Buchmann-Lenstra or Newton polygons
Quadratic fields
For quadratic fields the decomposition is decided entirely by the Kronecker symbol of the discriminant, giving an immediate answer with no factorisation at all — see quadratic field decomposition.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.8.1. 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.