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 those primes).

[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.