id summary reporter owner description type status priority milestone component resolution keywords cc merged author reviewer upstream work_issues branch commit dependencies stopgaps
28113 List of completely split primes is incomplete Stephan Ehlen "For number fields, the method `completely_split_primes`
may be incomplete.
Example:
{{{#!python
K. = QuadraticField(17)
K.completely_split_primes(20)
[13, 19]
}}}
However,
{{{#!python
K. = 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 them).
" defect new major sage-9.8 number fields number fields, splitting of primes Maarten Derickx Samuel Lelièvre Mckenzie West N/A