Orders, Ideals and Prime Decomposition
Ideals of the Maximal Order
Ideals and fractional ideals, unique factorisation into primes, and the group structure that makes the class group possible.
Engineering / MathematicsOrders, Ideals and Prime Decomposition2 min readKV-MATH-0584
The maximal order is a Dedekind domain, which means every non-zero ideal factors uniquely into prime ideals. This restores unique factorisation at the level of ideals even where it fails for elements.
Ideals and fractional ideals
- Integral ideal
- A module contained in the order and closed under multiplication by it.
- Fractional ideal
- A module with a denominator: some integer multiple of it is an integral ideal.
- Principal ideal
- Generated by a single element. The obstruction to all ideals being principal is the class group.
- Prime ideal
- A non-zero ideal whose quotient is an integral domain — for the maximal order, equivalently a field.
Unique factorisation
The ideal group
Non-zero fractional ideals form an abelian group under multiplication, free on the prime ideals. Principal ideals form a subgroup, and the quotient is the class group.
Inverses
Every non-zero fractional ideal is invertible in the maximal order. The inverse is computed as the set of field elements multiplying the ideal into the order, which is a module computation.
Norms and containment
The norm of an ideal is its index in the order, and it is multiplicative. Containment of ideals corresponds to divisibility: one ideal contains another exactly when it divides it.
Representation
Ideals are represented as modules in Hermite normal form or, more compactly, by two elements.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.6.2. 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.