Quotient rings should support function field construction if they are fields.

[BrentBaccala]

Univariate polynomial rings have associated function fields:

sage: R.<x> = QQ[]
sage: Frac(R)
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: Frac(R).function_field()
Rational function field in x over Rational Field

A similar construction should be possible for quotient rings, if they are fields:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: R.quo(y^5 - (x^3 + 2*x*y + 1/x)).function_field()
Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x

This ticket adds a function_field method to PolynomialQuotientRing_field

Component: algebra

Keywords: function field

Author: Brent Baccala

Branch/Commit: public/27011 @ 734fa82

Issue created by migration from https://trac.sagemath.org/ticket/27011

[BrentBaccala]

Branch: public/27011

[BrentBaccala]

Commit: 734fa82

[BrentBaccala]

Author: Brent Baccala

[BrentBaccala]

New commits:

734fa82

Trac #27011: add a function_field method to PolynomialQuotientRing_field

[embray]

comment:2

Retarging tickets optimistically to the next milestone. If you are responsible for this ticket (either its reporter or owner) and don't believe you are likely to complete this ticket before the next release (8.7) please retarget this ticket's milestone to sage-pending or sage-wishlist.

[embray]

comment:3

Ticket retargeted after milestone closed (if you don't believe this ticket is appropriate for the Sage 8.8 release please retarget manually)

[soehms]

comment:4

It's a good idea to have such a method! But I think it would be more useful if conversions between the both realizations would work.
Unfortunately, this is not the case:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L = R.quo(y^5 - (x^3 + 2*x*y + 1/x))
sage: F = L.function_field()
sage: F(L.an_element())
Traceback (most recent call last):
...
TypeError: cannot convert ybar/1 to an element of Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: L(F.an_element())
Traceback (most recent call last):
...
ValueError: y is not an element of the base field

It should not be difficult to improve this (for example via the _convert_map_from_ method), since both maps can be obtained as ring homomorphisms:

sage: f = L.hom(F.gens())
sage: g = F.hom(L.gens())
sage: f(L.an_element())
y
sage: g(F.an_element())
ybar
sage: (f*g).is_one()
True
sage: (g*f).is_one()
True

[embray]

comment:5

Moving tickets from the Sage 8.8 milestone that have been actively worked on in the last six months to the next release milestone (optimistically).

[embray]

comment:6

Ticket retargeted after milestone closed

[mkoeppe]

comment:7

Batch modifying tickets that will likely not be ready for 9.1, based on a review of the ticket title, branch/review status, and last modification date.

[seblabbe]

comment:8

changing the status to needs works according to the most recent comment

[mkoeppe]

comment:10

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[mkoeppe]

comment:11

Setting a new milestone for this ticket based on a cursory review.