Orders, Ideals and Prime Decomposition
Ideal Norm Computation
The norm of an ideal as its index in the order, its multiplicativity, and its use as a size measure and consistency check.
Engineering / MathematicsOrders, Ideals and Prime Decomposition2 min readKV-MATH-0588
The norm of an ideal is the number of residue classes modulo it. It is the primary size measure for ideals, it is multiplicative, and it is cheap to compute from the standard representation.
Definition and computation
Properties
- Multiplicative
- The norm of a product is the product of the norms. The single most useful property, and the basis of every consistency check on ideal arithmetic.
- Agrees with element norm
- For a principal ideal, the ideal norm is the absolute value of the element norm.
- Prime ideals
- A prime above a rational prime p has norm a power of p, the exponent being the residue degree.
- Finiteness
- Only finitely many ideals have norm below any bound, which is what makes factor bases finite.
The degree relation
Norms as a size measure
Reduction algorithms seek ideals of small norm in a given class, and factor bases consist of prime ideals of norm below a bound. In both cases the norm is the quantity being minimised or bounded — see factor base selection.
Smoothness
An ideal is smooth with respect to a factor base when its norm factors over the corresponding rational primes and each prime ideal appearing is in the base. Testing smoothness therefore begins with an integer factorisation of the norm — see smoothness.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.7.2. 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.