Coercions between Function Fields

[Julian Rüth]

The following are desired features.

Coercions between base fields should induce coercions on function fields:

sage: K.<x> = FunctionField(QQ)
sage: L.<x> = FunctionField(GaussianIntegers().fraction_field())
sage: L.has_coerce_map_from(K)
True

Also in towers:

sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^3 + 1)
sage: K.<x> = FunctionField(GaussianIntegers().fraction_field())
sage: R.<y> = K[]
sage: M.<y> = K.extension(y^3 + 1)
sage: M.has_coerce_map_from(L)
True

And when the base ring coerces into the extension:

sage: K.<x> = FunctionField(QQ)
sage: R.<I> = K[]
sage: L.<I> = K.extension(I^2 + 1)
sage: M.<x> = FunctionField(GaussianIntegers().fraction_field())
sage: M.has_coerce_map_from(L) # not tested, base_morphism is not implemented
True

[David Roe]

New commits:

​c227ee7

Adding more coercions between function fields and the ability to specify a morphism on the base field or constant field

[Julian Rüth]

Positive review. But one doctest fails because an is_injective is not implemented. I think that we should just disable the doctest because the failing doctest is not in the scope of this. (We discussed several ways of fixing this. The best would probably be to change the DefaultConvertMap to be a DefaultCoerceMap [does not exist yet] which is in the category of domain and codomain and would therefore by magically be injective.)

---

New commits:

​369a518

Remove obsolete not implemented comments

[David Roe]

I disabled the doctest. See #23184 for a followup which should allow the doctest to be re-enabled.

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New commits:

​b333178

Disable doctest for not-yet-working coercion

[Volker Braun]

Documentation doesn't build

[Volker Braun]

Was #23152...

[Volker Braun]

sage -t --long --warn-long 69.1 src/sage/categories/rings.py
**********************************************************************
File "src/sage/categories/rings.py", line 97, in sage.categories.rings.Rings.MorphismMethods.is_injective
Failed example:
    f = K.hom([codomain(1)]); f
Exception raised:
    Traceback (most recent call last):
      File "/mnt/disk/home/release/Sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 509, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/mnt/disk/home/release/Sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 872, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.categories.rings.Rings.MorphismMethods.is_injective[8]>", line 1, in <module>
        f = K.hom([codomain(Integer(1))]); f
      File "/mnt/disk/home/release/Sage/local/lib/python2.7/site-packages/sage/rings/function_field/function_field.py", line 1935, in hom
        raise ValueError("You must specify a morphism on the base field")
    ValueError: You must specify a morphism on the base field
**********************************************************************
File "src/sage/categories/rings.py", line 102, in sage.categories.rings.Rings.MorphismMethods.is_injective
Failed example:
    f.is_injective()
Expected:
    False
Got:
    True
**********************************************************************

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e2ad9b3

Fix zero morphism check

[Julian Rüth]

tests pass; david agreed to this fix.