Orders, Ideals and Prime Decomposition
Valuations and Uniformisers
Valuations at prime ideals, uniformising elements, and computing the exponent of a prime in an ideal factorisation.
Engineering / MathematicsOrders, Ideals and Prime Decomposition2 min readKV-MATH-0592
The valuation at a prime ideal records the exponent of that prime in a factorisation. Computing valuations and constructing elements with prescribed valuations are basic operations in ideal arithmetic and relation collection.
Valuation
- Uniformiser
- An element of valuation exactly one at the given prime. Exists for every prime ideal.
- Local ring
- The elements of non-negative valuation at a prime form a discrete valuation ring; the valuation is its length function.
- Product formula
- Valuations are additive on products, which is what makes them usable for factorisation bookkeeping.
Computing a valuation
Computing the valuation of an ideal at a prime
- Bound itThe valuation is bounded by the valuation of the norm divided by the residue degree, giving a quick upper limit.
- Divide repeatedlyMultiply by the inverse of the prime and test integrality, counting the successful steps.
- Or use a uniformiserMultiply by an element of valuation minus one at the prime and non-negative valuation elsewhere; count steps.
Constructing a uniformiser
A uniformiser is found by taking a random element of the prime and checking that it does not lie in the square of the prime. For a prime in two-element form, the second generator is usually already a uniformiser or becomes one after a small adjustment.
Factoring an ideal
Factoring an ideal into primes
- Compute the normFrom the Hermite representation.
- Factor the normAn integer factorisation.
- Decompose each primeFind the primes above each rational prime dividing the norm.
- Compute valuationsAt each candidate prime.
- VerifyThe product of prime norms to their valuations must equal the ideal norm.
Archimedean places
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.8. 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.