Orders, Ideals and Prime Decomposition
Essential Discriminant Divisors
Primes that divide the index for every choice of defining polynomial, why they exist, and what they force computationally.
Engineering / MathematicsOrders, Ideals and Prime Decomposition2 min readKV-MATH-0591
Some primes divide the index of the equation order no matter which defining polynomial is chosen. These essential discriminant divisors cannot be removed by reduction, and they force the general prime decomposition machinery.
The obstruction
The simple decomposition method requires the decomposition type above p to be realisable by a factorisation of a degree n polynomial over the field with p elements. When there are not enough irreducible polynomials of the needed degrees over that small field, no polynomial can realise the decomposition.
The smallest example
The classical case is a cubic field in which two splits completely into three primes of residue degree one. There are only two monic linear polynomials modulo two, so no cubic polynomial can factor into three distinct linear factors modulo two. Two is therefore an essential discriminant divisor for such a field.
Consequences
| Consequence | Detail |
|---|---|
| No power basis exists | The maximal order is not generated by powers of any single element |
| Reduction does not help | Polynomial reduction cannot remove the prime from the index |
| Simple decomposition fails | For that prime, permanently |
| General methods required | Buchmann-Lenstra or Newton polygons |
Which primes can be essential
Detection
An essential divisor is detected when the Dedekind criterion fails for a small prime and continues to fail after polynomial reduction. In practice the general decomposition method is simply applied to any prime for which the criterion fails, without distinguishing the cause.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.8.3. 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.