Opened 3 years ago

Last modified 2 months ago

#28113 new defect

List of completely split primes is incomplete — at Initial Version

Reported by: Stephan Ehlen Owned by:
Priority: major Milestone: sage-9.8
Component: number fields Keywords: number fields, splitting of primes
Cc: Maarten Derickx, Samuel Lelièvre, Mckenzie West Merged in:
Authors: Reviewers:
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: Stopgaps:

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Description

For number fields, the method completely_split_primes may be incomplete.

Example

  K.<a> = QuadraticField(17)
  K.completely_split_primes(20)
  [13, 19]

However,

  K.<a> = QuadraticField(17)
  K.ideal(2).factor()
  (Fractional ideal (-1/2*a - 3/2)) * (Fractional ideal (-1/2*a + 3/2))

The reason is that the factorization of the defining polynomial mod p does not always give the correct answer. It does in all but finitely many cases, the exception being primes that divide the index of ZZ[a] in the ring of integers of K.

A possible solution would be to use the function K.ideal(p).factor() and determine the length of the splitting (at least for those finitely many primes in case we can easily determine those primes).

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