List of completely split primes is incomplete

[ehlen]

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 them).

[embray]

Ticket retargeted after milestone closed

[mkoeppe]

Moving tickets to milestone sage-9.2 based on a review of last modification date, branch status, and severity.

[mkoeppe]

Moving to 9.4, as 9.3 has been released.

[slelievre]

Reported again at #32982. The example there might serve as a doctest here.

[dimpase]

CC-ing the code author