Opened 2 years ago

Last modified 2 years ago

## #25836 new defect

# in PrincipalIdealDomains() and in UniqueFactorizationDomains() give wrong answers

Reported by: | ababei | Owned by: | |
---|---|---|---|

Priority: | major | Milestone: | sage-8.3 |

Component: | commutative algebra | Keywords: | Days95 |

Cc: | vdelecroix | Merged in: | |

Authors: | Reviewers: | ||

Report Upstream: | N/A | Work issues: | |

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description

One of the tests in `src/sage/categories/principal_ideal_domains.py`

(lines 80-83) is testing that the following ring is not a UFD

sage: K = QuadraticField(5) sage: O = K.maximal_order() sage: O in UniqueFactorizationDomains() False

However, trying rings of integers of number fields, which are UFDs or PIDs still gives false as an answer.

sage: K = QuadraticField(-1) sage: O = K.maximal_order() sage: O in UniqueFactorizationDomains() False sage: O in PrincipalIdealDomains() False

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That's an interesting example, since the integer ring of

`Q(\sqrt(5))`

is actually a UFD and a PID.Looking at the code of

`UniqueFactorizationDomains().__contains__`

one can see that any non-trivial functionality should go into O.is_unique_factorization_domain(). In this case the implementation could be`return self.class_number() == 1`

. However, such a method isn't there, so you're seeing a true/false based on the stored categorical info, and integer rings are not categorically UFDs.There may be bad ramifications from answering

`UniqueFactorizationDomains().__contains__`

via a probably very expensive class group computation, so there might be good practical reasons why a more functional`is_unique_factorization_domain`

method isn't provided.In any case, it may be worth changing the example to

`K=QuadraticField(-5)`

or so. It looks a bit silly to use a mathematically UFD in a setting to illustrate non-UFD behaviour.I also don't see how

`PolynomialRing(O,'x')`

can be a PID: the ideal`(7,x)`

is really not principal. Were these doctests written by a non-mathematician?