Maximal Orders and Decomposition II
The Pohst-Zassenhaus Theorem
The theorem underlying maximal order computation: enlarging an order by the ring of multipliers of its radical.
Engineering / MathematicsMaximal Orders and Decomposition II2 min readKV-MATH-0611
The Pohst-Zassenhaus theorem tells you how to enlarge a non-maximal order and, crucially, when to stop. It is the theoretical engine of the Round 2 algorithm.
The statement
For an order and a prime, consider the radical of the order modulo that prime, and the ring of elements of the field multiplying that radical into itself. That ring is an order containing the original, and it equals the original exactly when the original is maximal at the prime.
The p-radical
- Definition
- The set of elements of the order whose image in the quotient modulo p is nilpotent.
- Alternative description
- The intersection of the prime ideals of the order above p.
- Computation
- As the kernel of an iterated Frobenius-like map — see radical computation.
Termination
Each enlargement strictly reduces the power of the prime dividing the index, so the number of steps is bounded by the exponent of that prime in the polynomial discriminant.
Why the radical
Relation to the algorithm
The theorem is applied prime by prime inside the Round 2 algorithm, with the Dedekind criterion used first as a cheap test to avoid the enlargement work where it is unnecessary.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.1.1. 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.