Opened 7 years ago

Last modified 7 years ago

## #15865 new defect

# Should there be a method on a rational function field that returns the ring it came from?

Reported by: | darij | Owned by: | |
---|---|---|---|

Priority: | major | Milestone: | |

Component: | algebra | Keywords: | polynomials, fraction field, categories |

Cc: | tscrim, sage-combinat | Merged in: | |

Authors: | Reviewers: | ||

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

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description

If I have a fraction field, how do I find the ring whose fraction field it is? Note that, since a fraction field sometimes serves several base rings at the same time, this can mean:

- the base ring from which the fraction field was constructed (possibly thread-unsafe?);

- a "canonical" base ring for which the fraction field can be constructed;

- or anything inbetween.

I'm not sure which of these are feasible; I'd be happy with a method that returns me a polynomial ring if I apply it to the fraction field of said polynomial ring. There is the `_base`

attribute which seems to give the base ring, but I'd prefer an exposed method.

I assume this also does the trick:

sage: g = FractionField(PolynomialRing(QQ, ['x'])) sage: parent(g.zero().numerator()) Rational Field

but it feels like a hack...

### Change History (3)

### comment:1 Changed 7 years ago by

### comment:2 Changed 7 years ago by

Thanks! What about adding some doc like this:

Return the base of ``self``. This means a reasonable choice of a ring `R` such that ``self`` is the fraction field of ``self``. For instance, if ``self`` is the fraction field of a polynomial ring, then ``self.base()`` is said polynomial ring (as opposed to ``self.base_ring()``, which is the base ring over which the polynomial ring is defined). .. WARNING:: This might not be the ring *you* used to construct ``self``. For instance: sage: QQ.base() # not ZZ Rational Field sage: Frac(Frac(PolynomialRing(QQ, 'x'))).base() Univariate Polynomial Ring in x over Rational Field

(Note that it is not the parent of self.zero().numerator()...)

### comment:3 Changed 7 years ago by

Hmm....I think we maybe should have a more clearly named method such as `fraction_field_base()`

for all fraction fields. Oh also I remembered:

sage: g = FractionField(QQ['x']) sage: g.construction() (FractionField, Univariate Polynomial Ring in x over Rational Field)

Although on another ticket we decided to have the fraction field of Laurent polynomials be the fraction field of usual polynomials (I forget the number currently, but I can find it if you want). I think this is the right thing to do FTR. Hence we should not expect to have `FF(R).base() == R`

in general. Moreover, I think `QQ.fraction_field_base()`

should be `ZZ`

. Although perhaps not `base()`

, but that would definitely need a sage-devel discussion.

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There's the somewhat unclearly named

`base()`

method: