Changes between Initial Version and Version 7 of Ticket #28113


Ignore:
Timestamp:
Dec 7, 2021, 2:55:01 PM (10 months ago)
Author:
Samuel Lelièvre
Comment:

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

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  • Ticket #28113

    • Property Cc Maarten Derickx Samuel Lelièvre added
    • Property Milestone changed from sage-8.9 to sage-9.5
  • Ticket #28113 – Description

    initial v7  
    1 For number fields, the method {{{completely_split_primes}}} may be incomplete.
     1For number fields, the method `completely_split_primes`
     2may be incomplete.
    23
    3 == Example ==
     4Example:
     5
    46{{{#!python
    5   K.<a> = QuadraticField(17)
    6   K.completely_split_primes(20)
    7   [13, 19]
     7K.<a> = QuadraticField(17)
     8K.completely_split_primes(20)
     9[13, 19]
    810}}}
    911
    1012However,
    1113{{{#!python
    12   K.<a> = QuadraticField(17)
    13   K.ideal(2).factor()
    14   (Fractional ideal (-1/2*a - 3/2)) * (Fractional ideal (-1/2*a + 3/2))
     14K.<a> = QuadraticField(17)
     15K.ideal(2).factor()
     16(Fractional ideal (-1/2*a - 3/2)) * (Fractional ideal (-1/2*a + 3/2))
    1517}}}
    1618
    17 The reason is that the factorization of the defining polynomial mod p does
    18 not always give the correct answer.
    19 It does in all but finitely many cases, the exception being primes that divide
    20 the index of ZZ[a] in the ring of integers of K.
     19The reason is that the factorization of the defining polynomial
     20mod p does  not always give the correct answer.
     21It does in all but finitely many cases, the exception
     22being primes that divide the index of `ZZ[a]`
     23in the ring of integers of `K`.
    2124
    2225A possible solution would be to use the function
    23 {{{K.ideal(p).factor()}}} and determine the length
    24 of the splitting (at least for those finitely many primes
    25 in case we can easily determine those primes).
     26`K.ideal(p).factor()` and determine the length of
     27the splitting (at least for those finitely many
     28primes in case we can easily determine them).
    2629