Changes between Initial Version and Version 7 of Ticket #28113
 Timestamp:
 Dec 7, 2021, 2:55:01 PM (10 months ago)
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Ticket #28113
 Property Cc Maarten Derickx Samuel Lelièvre added

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changed from
sage8.9
tosage9.5

Ticket #28113 – Description
initial v7 1 For number fields, the method {{{completely_split_primes}}} may be incomplete. 1 For number fields, the method `completely_split_primes` 2 may be incomplete. 2 3 3 == Example == 4 Example: 5 4 6 {{{#!python 5 6 7 7 K.<a> = QuadraticField(17) 8 K.completely_split_primes(20) 9 [13, 19] 8 10 }}} 9 11 10 12 However, 11 13 {{{#!python 12 13 14 14 K.<a> = QuadraticField(17) 15 K.ideal(2).factor() 16 (Fractional ideal (1/2*a  3/2)) * (Fractional ideal (1/2*a + 3/2)) 15 17 }}} 16 18 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. 19 The reason is that the factorization of the defining polynomial 20 mod p does not always give the correct answer. 21 It does in all but finitely many cases, the exception 22 being primes that divide the index of `ZZ[a]` 23 in the ring of integers of `K`. 21 24 22 25 A 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 27 the splitting (at least for those finitely many 28 primes in case we can easily determine them). 26 29