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: |

### 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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