number_field_elements_from_algebraics should create embedded number field elements

[mkoeppe]

This is a follow-up on #17830 and #18819.

#17830 made this possible:

sage: x = polygen(ZZ)
sage: K.<cbrt2> = NumberField(x^3 - 2, embedding=AA.polynomial_root(x^3-2, RIF(0,3)))
sage: 6064/4813 < cbrt2 < 90325/71691
True

But if one uses number_field_elements_from_algebraics, one gets only non-embedded number field elements:

sage: x = polygen(ZZ)
sage: K, cbrt2, hom = number_field_elements_from_algebraics(AA.polynomial_root(x^3-2, RIF(0,3)))
sage: 6064/4813 < cbrt2 < 90325/71691
False            # (an perhaps an error in Python 3?)

This ticket adds an option embedded to create embedded number field elements.

sage: x = polygen(ZZ)
sage: K, cbrt2, hom = number_field_elements_from_algebraics(AA.polynomial_root(x^3-2, RIF(0,3)), embedded=True)
sage: 6064/4813 < cbrt2 < 90325/71691
True

TODECIDE:

• Close ticket #5355?

[mkoeppe]

While looking at related tickets, I noticed the very old ticket #5355, which should be closed I think.

[jipilab]

Here is a first version. Comments are welcome.

One question is whether RealIntervalField is the right choice?

---

New commits:

​26dc3e4

Added option for embedded NumberField

[jipilab]

Maybe it would be nice to also add the optional parameter to the method as_number_field_element as well...

[jipilab]

The following should work but it does not:

sage: UCF = UniversalCyclotomicField()
sage: E = UCF.gen(5)
sage: my_nums = [-52*E - 136*E^2 - 136*E^3 - 52*E^4, -66*E - 168*E^2 - 168*E^3 - 66*E^4, -46*E - 114*E^2 - 114*E^3 - 46*E^4, -24*E - 58*E^2 - 58*E
....: ^3 - 24*E^4, sqrt(2)]
sage: [n(_) for _ in my_nums]
[187.914855054991 - 8.67361737988404e-18*I,
 231.039466852489 - 1.04083408558608e-17*I,
 156.026311234993 - 8.67361737988404e-18*I,
 79.0131556174964 - 4.33680868994202e-18*I,
 1.41421356237310]
sage: res = number_field_elements_from_algebraics(my_nums,embedded=True)
Traceback (most recent call last):
...
TypeError: unable to convert -0.4370160244488211? - 1.344997023927915?*I to real interval

The choice of root shoould always be made real...

[git]

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

​db778ee

Made it possible to deal with complex number fields

[jipilab]

It is now possible to get this:

sage: UCF = UniversalCyclotomicField()
sage: E = UCF.gen(5)
sage: L.<b> = NumberField(x^2-189*x+16, embedding=200)
sage: my_nums = [-52*E - 136*E^2 - 136*E^3 - 52*E^4, L.gen()._algebraic_(AA),sqrt(2)]
sage: aa_my_nums = [AA(_) for _ in my_nums]
sage: res = number_field_elements_from_algebraics(aa_my_nums,embedded=True)
sage: res
(Number Field in a with defining polynomial y^8 - 35670*y^6 + 476899047*y^4 - 2832410271650*y^2 + 6305298701739921,
 [2310/26212773509*a^7 - 185432947/78638320527*a^5 + 1652517502195/78638320527*a^3 - 4904676315215467/78638320527*a + 94,
  -1238/2803377488467023*a^7 + 185460719/11213509953868092*a^5 - 2754936849443/11213509953868092*a^3 + 8180694680816975/3737836651289364*a + 189/2,
  -1979/1887160880826*a^7 + 26472586/943580440413*a^5 - 235822245043/943580440413*a^3 + 466325019915415/629053626942*a],
 Ring morphism:
   From: Number Field in a with defining polynomial y^8 - 35670*y^6 + 476899047*y^4 - 2832410271650*y^2 + 6305298701739921
   To:   Algebraic Real Field
   Defn: a |--> 96.9475535136628?)
sage: res[0].gen_embedding()
96.9475535136628?
sage: res = number_field_elements_from_algebraics(aa_my_nums,embedded=False)
sage: res[0].gen_embedding()  # Nothing is returned(!?!?!)
sage:

Perhaps it would be nice to say that there is no embedding instead of returning nothing?

[mkoeppe]

The returned morphism is not able to convert the elements.

sage: res[2](my_nums[0])
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-8-74370e91d982> in <module>()
----> 1 res[Integer(2)](my_nums[Integer(0)])

/Users/mkoeppe/s/sage/some-sage-install-prefix/lib/python2.7/site-packages/sage/categories/map.pyx in sage.categories.map.Map.__call__ (build/cythonized/sage/categories/map.c:6978)()
    781                 x = D(x)
    782             except (TypeError, NotImplementedError):
--> 783                 raise TypeError("%s fails to convert into the map's domain %s, but a `pushforward` method is not properly implemented" % (x, D))
    784         else:
    785             x = converter(x)

TypeError: -52*E(5) - 136*E(5)^2 - 136*E(5)^3 - 52*E(5)^4 fails to convert into the map's domain Number Field in a with defining polynomial y^8 - 35670*y^6 + 476899047*y^4 - 2832410271650*y^2 + 6305298701739921, but a `pushforward` method is not properly implemented

Edit: I withdraw this comment; I misremembered what hom is supposed to do.

[mmezzarobba]

Two quick comments:

• you may want to pathc as_number_field_element() too,
• don't forget to add tests covering the case where some or all of the elements are rationals or elements of quadratic extensions (rationals, quadratic number field elements and higher-degree number field elements don't have exactly the same interface...)

[jipilab]

Replying to mmezzarobba:

Two quick comments:

• you may want to pathc as_number_field_element() too,
• don't forget to add tests covering the case where some or all of the elements are rationals or elements of quadratic extensions (rationals, quadratic number field elements and higher-degree number field elements don't have exactly the same interface...)

Yes, that sounds very reasonable. I will do that, along with fixing what mkoeppe mentionned above.

[mkoeppe]

Doesn't always seem to give real embeddings:

sage: x = polygen(ZZ); elts = number_field_elements_from_algebraics([sqrt(2), AA.polynomial_root(x^3-2, RIF(0,3))], embedded=True)[1]
sage: [AA(y) for y in elts]
...
TypeError: unable to convert -0.561231024154687? - 0.9720806486198328?*I to real interval

[jipilab]

Replying to mkoeppe:

The returned morphism is not able to convert the elements.

sage: res[2](my_nums[0])
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-8-74370e91d982> in <module>()
----> 1 res[Integer(2)](my_nums[Integer(0)])

/Users/mkoeppe/s/sage/some-sage-install-prefix/lib/python2.7/site-packages/sage/categories/map.pyx in sage.categories.map.Map.__call__ (build/cythonized/sage/categories/map.c:6978)()
    781                 x = D(x)
    782             except (TypeError, NotImplementedError):
--> 783                 raise TypeError("%s fails to convert into the map's domain %s, but a `pushforward` method is not properly implemented" % (x, D))
    784         else:
    785             x = converter(x)

TypeError: -52*E(5) - 136*E(5)^2 - 136*E(5)^3 - 52*E(5)^4 fails to convert into the map's domain Number Field in a with defining polynomial y^8 - 35670*y^6 + 476899047*y^4 - 2832410271650*y^2 + 6305298701739921, but a `pushforward` method is not properly implemented

The homomorphism goes the other way around... Thanks to Moritz for noticing that.

[git]

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

​a9045bf

Added tests and made them pass

[jipilab]

Okay. I added some tests and now it seems to work.

With the current commit, I have one failing doctest for which I do not know what to do:

File "qqbar.py", line 2098, in sage.rings.qqbar.number_field_elements_from_algebraics
Failed example:
    number_field_elements_from_algebraics(rt2c)
Expected:
    (Number Field in a with defining polynomial y^4 + 2*y^2 + 4, 1/2*a^3, Ring morphism:
        From: Number Field in a with defining polynomial y^4 + 2*y^2 + 4
        To:   Algebraic Field
        Defn: a |--> -0.7071067811865475? - 1.224744871391589?*I)
Got:
    (Number Field in a with defining polynomial y^2 - 2, a, Ring morphism:
       From: Number Field in a with defining polynomial y^2 - 2
       To:   Algebraic Real Field
       Defn: a |--> 1.414213562373095?)

This is in the tests section and the explanation given is exactly a situation that we have to outsmart when we want to be able to accept cyclotomic fields elements to be parsed to real embedded number fields.

It seems to me that the proposed code improves the situation as Sage now recognize that it is a real number... I can not see a reason right now to go against that. Am I missing something?

[dimpase]

Modifying the test in comment 19 looks OK to me, but perhaps John can confirm I'm not missing anything...

[chapoton]

You should just go on and fix the doctest, I think.

[mkoeppe]

as_number_field_element needs to pass on its new arguments, and corresponding doctests should be added.

[git]

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

​60c527f

Merge remote-tracking branch 'trac/develop' into t/20181/public/algnum_to_numberfield

[git]

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

​f1627e7

AlgebraicNumber_base.as_number_field_element: Pass new kwargs to number_field_elements_from_algebraics

[mkoeppe]

... still needs a corresponding doctest

[git]

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

​d7e31a2	AlgebraicNumber_base.as_number_field_element: Add doctest
​284c217	Merge remote-tracking branch 'trac/develop' into t/20181/public/algnum_to_numberfield
​41b1142	AlgebraicNumber_base.as_number_field_element: Add doctests (currently failing)

[mkoeppe]

I've added some doctests, but the implementation needs more work as there are failures.

[git]

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

​2eb3320

number_field_elements_from_algebraics: Mark a random doctest # random

[mkoeppe]

Replying to dimpase:

Modifying the test in comment 19 looks OK to me, but perhaps John can confirm I'm not missing anything...

The context of this doctest suggested that it should be marked # random, which I did in 2eb3320.

[tscrim]

Ready for re-review?

Some quick comments. It is considered a bad practice to have bare except: statements. It is probably sufficient to catch ValueError and TypeError. Also, the standard doc format for INPUT: would be

    - ``numbers`` -- a number or list of numbers

    - ``minimal`` -- boolean (default: ``False``); whether to minimize the
      degree of the extension

    - ``same_field`` -- boolean (default: ``False``); see below

    - ``embedded`` -- boolean (default: ``False``); whether to make the
      :func:`NumberField` embedded

    - ``prec`` -- integer (default: ``53``); the number of bit of precision for
      the embedding

        - ``minimal`` -- boolean (default: ``False``); whether to minimize the
          degree of the extension

        - ``embedded`` -- boolean (default: ``False``); whether to make the
          :func:`NumberField` embedded
    
        - ``prec`` -- integer (default: ``53``); the number of bit of precision for
          the embedding

[mkoeppe]

Not ready for review; this needs more work -- see the failing doctests that I added in 41b1142.

[mkoeppe]

Replying to mkoeppe:

Not ready for review; this needs more work -- see the failing doctests that I added in 41b1142.

For reference: These are the doctest failures.

$ sage -t src/sage/rings/qqbar.py 
too many failed tests, not using stored timings
Running doctests with ID 2018-09-27-16-59-10-65572ec7.
Using --optional=4ti2,bliss,ccache,dochtml,gdb,latte_int,lidia,lrslib,mpir,normaliz,notedown,pandoc_attributes,pynormaliz,pyqnormaliz,python2,rst2ipynb,sage
Doctesting 1 file.
sage -t src/sage/rings/qqbar.py
**********************************************************************
File "src/sage/rings/qqbar.py", line 3696, in sage.rings.qqbar.AlgebraicNumber_base.as_number_field_element
Failed example:
    elt == rt
Expected:
    True
Got:
    False
**********************************************************************
File "src/sage/rings/qqbar.py", line 3698, in sage.rings.qqbar.AlgebraicNumber_base.as_number_field_element
Failed example:
    AA(elt)
Exception raised:
    Traceback (most recent call last):
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 659, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 1070, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.rings.qqbar.AlgebraicNumber_base.as_number_field_element[12]>", line 1, in <module>
        AA(elt)
      File "sage/structure/parent.pyx", line 921, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9679)
        return mor._call_(x)
      File "sage/structure/coerce_maps.pyx", line 145, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4574)
        raise
      File "sage/structure/coerce_maps.pyx", line 140, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4442)
        return C._element_constructor(x)
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/rings/qqbar.py", line 738, in _element_constructor_
        return x._algebraic_(AA)
      File "sage/rings/number_field/number_field_element.pyx", line 2709, in sage.rings.number_field.number_field_element.NumberFieldElement._algebraic_ (build/cythonized/sage/rings/number_field/number_field_element.cpp:26930)
        emb = refine_embedding(emb, infinity)
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/rings/number_field/number_field.py", line 11329, in refine_embedding
        diffs = [(RC(ee(K.gen()))-old_root).abs() for ee in elist]
      File "sage/rings/real_mpfi.pyx", line 699, in sage.rings.real_mpfi.RealIntervalField_class.__call__ (build/cythonized/sage/rings/real_mpfi.c:6094)
        return Parent.__call__(self, x)
      File "sage/structure/parent.pyx", line 921, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9679)
        return mor._call_(x)
      File "sage/structure/coerce_maps.pyx", line 271, in sage.structure.coerce_maps.NamedConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:5999)
        cdef Element e = method(C)
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/rings/qqbar.py", line 4007, in interval
        raise TypeError("unable to convert {} to real interval".format(self))
    TypeError: unable to convert -2.618826569900552? - 0.2237500597843041?*I to real interval
**********************************************************************
File "src/sage/rings/qqbar.py", line 3700, in sage.rings.qqbar.AlgebraicNumber_base.as_number_field_element
Failed example:
    RR(elt)
Exception raised:
    Traceback (most recent call last):
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 659, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 1070, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.rings.qqbar.AlgebraicNumber_base.as_number_field_element[13]>", line 1, in <module>
        RR(elt)
      File "sage/structure/parent.pyx", line 921, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9679)
        return mor._call_(x)
      File "sage/structure/coerce_maps.pyx", line 271, in sage.structure.coerce_maps.NamedConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:5993)
        cdef Element e = method(C)
      File "sage/rings/number_field/number_field_element.pyx", line 1802, in sage.rings.number_field.number_field_element.NumberFieldElement._mpfr_ (build/cythonized/sage/rings/number_field/number_field_element.cpp:19625)
        return R(R.complex_field()(self))
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/rings/complex_field.py", line 381, in __call__
        return Parent.__call__(self, x)
      File "sage/structure/parent.pyx", line 921, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9679)
        return mor._call_(x)
      File "sage/structure/coerce_maps.pyx", line 145, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4574)
        raise
      File "sage/structure/coerce_maps.pyx", line 140, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4442)
        return C._element_constructor(x)
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/rings/complex_field.py", line 423, in _element_constructor_
        return ComplexNumber(self, x)
      File "sage/rings/complex_number.pyx", line 200, in sage.rings.complex_number.ComplexNumber.__init__ (build/cythonized/sage/rings/complex_number.c:4535)
        rr = R(real)
      File "sage/structure/parent.pyx", line 921, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9679)
        return mor._call_(x)
      File "sage/structure/coerce_maps.pyx", line 271, in sage.structure.coerce_maps.NamedConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:5993)
        cdef Element e = method(C)
      File "sage/rings/number_field/number_field_element.pyx", line 1802, in sage.rings.number_field.number_field_element.NumberFieldElement._mpfr_ (build/cythonized/sage/rings/number_field/number_field_element.cpp:19625)
        return R(R.complex_field()(self))
      File "/Users/mkoeppe/s/sage/sage-rebasing/worktree-clean/local/lib/python2.7/site-packages/sage/rings/complex_field.py", line 381, in __call__
        return Parent.__call__(self, x)
      File "sage/structure/parent.pyx", line 921, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9679)
        return mor._call_(x)
      File "sage/structure/coerce_maps.pyx", line 145, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4574)
        raise
      File "sage/structure/coerce_maps.pyx", line 140, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4442)
.....................
      File "sage/categories/map.pyx", line 176, in sage.categories.map.Map.__copy__ (build/cythonized/sage/categories/map.c:4133)
        cdef Map out = Element.__copy__(self)
      File "sage/structure/element.pyx", line 605, in sage.structure.element.Element.__copy__ (build/cythonized/sage/structure/element.c:5273)
        D = self.__dict__
      File "sage/structure/element.pyx", line 493, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4550)
        return self.getattr_from_category(name)
      File "sage/structure/element.pyx", line 506, in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4659)
        return getattr_from_other_class(self, cls, name)
      File "sage/cpython/getattr.pyx", line 386, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2428)
        if isinstance(self, cls):
    RuntimeError: maximum recursion depth exceeded while calling a Python object
**********************************************************************
1 item had failures:
   3 of  21 in sage.rings.qqbar.AlgebraicNumber_base.as_number_field_element
    [1438 tests, 3 failures, 14.69 s]
----------------------------------------------------------------------
sage -t src/sage/rings/qqbar.py  # 3 doctests failed
----------------------------------------------------------------------

For the infinite recursion in the last failure, perhaps could someone experienced with the Sage coercion system help?

[tscrim]

So I think I know what is going wrong. The issue is that there is no coercion from RIF -> RR; it goes the other way around. So with these changes, I can avoid the infinite recursion with RR(elt), but I get other failures:

src/sage/rings/qqbar.py

@@ -2197 +2197 @@
 
     # Make the numbers have a real root
     real_numbers = []
+    from sage.rings.real_mpfr import RealField
     for v in numbers:
         if v._exact_field().is_complex() and real_case:
             # the number comes from a complex algebraic number field
-            embedded_rt = v.interval_fast(RealIntervalField(prec))
+            embedded_rt = v.interval_fast(RealField(prec))
             root = ANRoot(v.minpoly(), embedded_rt)
             real_nf = NumberField(v.minpoly(),'a')
             new_ef = AlgebraicGenerator(real_nf, root)
@@ -2223 +2224 @@
     if fld is not QQ and embedded:
         assert real_case
         exact_generator = hom(fld.gen(0))
-        embedding_field = RealIntervalField(prec)
+        embedding_field = RealField(prec)
         embedded_gen = embedding_field(exact_generator)
         embedded_field = NumberField(fld.defining_polynomial(),fld.variable_name(),embedding=embedded_gen)
-        
         inter_hom = fld.hom([embedded_field.gen(0)])
         nums = map(inter_hom, nums)
         hom = embedded_field.hom([gen.root_as_algebraic()])

Note that with your current branch, the replacement of the RR(elt) by RIF(elt) works.

The reason you get the infinite recursion is somewhere in the number field coercion (conversion?)( code, it is trying to construct a complex number, which then tries to create the RR, but it cannot do that, so it tried to construct a number field element, and hence the cycle. Or something like that. The reason it normally doesn't cycle is because it separately knows how to get the the information it needs (if it even falls into this cycle). I don't completely understand the mechanism, but this is what the traceback is suggesting to me.

[mmezzarobba]

Replying to tscrim:

So I think I know what is going wrong. The issue is that there is no coercion from RIF -> RR; it goes the other way around.

Note that #15114 is for removing this coercion, but is currently blocked by more or less the same issues with comparisons between elements of different parents that need to be solved for the Python 3 transition.

[jipilab]

Just as a summary here are things that need to be done (maybe some of them are already done):

• you may want to patch as_number_field_element() too,
• don't forget to add tests covering the case where some or all of the elements are rationals or elements of quadratic extensions (rationals, quadratic number field elements and higher-degree number field elements don't have exactly the same interface...)
• Fix doctest in File "qqbar.py", line 2098, in sage.rings.qqbar.number_field_elements_from_algebraics
• as_number_field_element needs to pass on its new arguments, and corresponding doctests should be added.
• Fix Comment 34 and Comment 36.

[git]

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

​5793bcf

Merge branch 'public/algnum_to_numberfield' of trac.sagemath.org:sage into 20181

[jipilab]

New commits:

​5793bcf

Merge branch 'public/algnum_to_numberfield' of trac.sagemath.org:sage into 20181

---

New commits:

​5793bcf

Merge branch 'public/algnum_to_numberfield' of trac.sagemath.org:sage into 20181

[jipilab]

While rereading the docstring, I realized that it is quite confusing. Especially the following lines:

2154     Note that for the first example, where \sage does not realize that 
2155     the number is real, we get a homomorphism to ``QQbar``; but with
2156     ``minimal=True``, we get a homomorphism to ``AA``.  If we specify
2157     both ``minimal=True`` and ``same_field=True``, we get a second
2158     degree extension (minimal) that maps back to ``QQbar``.

[git]

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

​cb63fe5

Added some doctest for embedded case

[git]

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

​b913dcc

Fixed usage of RealField and docstring

[git]

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

​b153ef7

fixed docstring

[git]

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

​61e4df9

Fix docstring even more

[jipilab]

I deleted an old comment that should have been deleted in 2010 from ticket #9343.

It now looks ready for review.

[git]

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

​9255f2c

pyflakes

[mkoeppe]

Looks good to me and works well within #25097. But it would be good if another reviewer could take a look as well.

[dimpase]

It would be great is a semi-full-time number theorist looks into this...

[cremona]

I almost never use embedded number fields (instead I make explicit embeddings into RR or CC as needed), but this all looks reasonable and there is a lot of new documentation and examples, which is good. So OK from me.

[chapoton]

you used "map" in a way that is not compatible with py3, please review the fix in #27767