repr of NumberFields (the parents) should indicate its embedding if there is one

[mkoeppe]

As discussed in #21105, number fields with coercion embeddings, in particular with real embeddings, behave quite differently from those without - but there's no indication of embeddings in the print representation:

sage: K.<a> = NumberField(x^2 - 2)
sage: a.parent()
Number Field in a with defining polynomial x^2 - 2
sage: K.<sqrt2> = NumberField(x^2 - 2, embedding=1.4)
sage: sqrt2.parent()
Number Field in sqrt2 with defining polynomial x^2 - 2

This ticket changes the print representation when there is an embedding as follows.

Number Field in a with defining polynomial x^13 - 2/3*x + 3 with a = -1.106745229567614?

This also works well for more complicated situations such as this one:

            sage: K.<a> = NumberField(x^4 - 3); K
            Number Field in a with defining polynomial x^4 - 3
            sage: H.<b>, from_H = K.subfield(a^2)
            sage: H
            Number Field in b with defining polynomial x^2 - 3 with b = a^2

[vdelecroix]

Shorter sentence would be better.

Real Number Field in sqrt2 as the root of x^2 - 2 in [1.41, 1.42]

Ideally (as above), the X in near X should have the form of an interval [1.41,1.42] that specifies uniquely the root.

[vdelecroix]

Or possibly

Real Number Field in sqrt2=1.41421356237309? with defining polynomial x^2 - 2

[mkoeppe]

AlgebraicGenerator (from qqbar) uses

   Number Field in a with defining polynomial y^2 - y - 1 with a in 1.618033988749895?

[vdelecroix]

Replying to mkoeppe:

AlgebraicGenerator (from qqbar) uses

   Number Field in a with defining polynomial y^2 - y - 1 with a in 1.618033988749895?

Which is not that bad except the a in 1.618033988749895?. Would make more sense with a in [1.61, 1.62] or a = 1.618033988749895?.

[nbruin]

This should work for number fields with a real embedding as well as a complex one. The interval notation is painful for the latter, so "with a=..." would probably work better. Also good if we end up with number fields with a p-adic embedding specified:

K.<a>=NumberField(x^2+1,embedding=pAdicField(5)(-1).sqrt())

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​d40c9e6

NumberField_generic._repr_: Print embedding if there is one

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​ba3b35d

NumberField_generic._repr_: Print embedding if there is one

[mkoeppe]

In this patch I am using a format with with a = ... now (as suggested by nbruin), which works well also for the complex case:

Number Field in a with defining polynomial x^13 - 2/3*x + 3 with a = -1.106745229567614?

and for more complicated situations such as this one:

            sage: K.<a> = NumberField(x^4 - 3); K
            Number Field in a with defining polynomial x^4 - 3
            sage: H.<b>, from_H = K.subfield(a^2)
            sage: H
            Number Field in b with defining polynomial x^2 - 3 with b = a^2

The output for the following looks a bit strange because printing goes through sage.rings.real_lazy.LazyBinop:

sage: QuadraticField(-1, 'a')
Number Field in a with defining polynomial x^2 + 1 with a = 1*I

Should we special case this?

[git]

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

​ed3aab0

Update doctest outputs

[jipilab]

The patchbot gave 2 pyflakes warnings, I repaired one of them, the other one I can not make sure that it does not have side effects.

Otherwise, the patchbot is happy and the resolution looks reasonable. I set it as positive review, if you agree @mkoeppe.

---

New commits:

​875731c

fixed one pyflakes

[jipilab]

Whoooops!

[git]

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

​895b962

Made doctests pass

[jipilab]

Let's see if the bot finds more. I hope I did not forget any...

[jipilab]

Failing doctests

[git]

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

​353b01b

Make more doctests pass

[git]

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

​2ee8fcc

Make more doctests pass

[jipilab]

sage -t --long src/sage/modular/local_comp/smoothchar.py
**********************************************************************
File "src/sage/modular/local_comp/smoothchar.py", line 461, in sage.modular.local_comp.smoothchar.SmoothCharacterGroupGeneric._coerce_map_from_
Failed example:
    G.coerce(GK.character(0, [4]))
Exception raised:
    Traceback (most recent call last):
      File "/home/sage-patchbot/sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 671, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/home/sage-patchbot/sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 1095, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.modular.local_comp.smoothchar.SmoothCharacterGroupGeneric._coerce_map_from_[7]>", line 1, in <module>
        G.coerce(GK.character(Integer(0), [Integer(4)]))
      File "sage/structure/parent.pyx", line 1114, in sage.structure.parent.Parent.coerce (build/cythonized/sage/structure/parent.c:10541)
        cpdef coerce(self, x):
      File "sage/structure/parent.pyx", line 1144, in sage.structure.parent.Parent.coerce (build/cythonized/sage/structure/parent.c:10470)
        raise TypeError(_LazyString(_lazy_format, ("no canonical coercion from %s to %s", parent(x), self), {}))
    TypeError: no canonical coercion from Group of smooth characters of Q_3* with values in Number Field in i with defining polynomial x^2 + 1 with i = 1*I to Group of smooth characters of Q_3* with values in Rational Field
**********************************************************************
1 item had failures:
   1 of  14 in sage.modular.local_comp.smoothchar.SmoothCharacterGroupGeneric._coerce_map_from_
    [303 tests, 1 failure, 2.58 s]
sage -t --long src/sage/tests/books/computational-mathematics-with-sagemath/polynomes_doctest.py
**********************************************************************
File "src/sage/tests/books/computational-mathematics-with-sagemath/polynomes_doctest.py", line 177, in sage.tests.books.computational-mathematics-with-sagemath.polynomes_doctest
Failed example:
    for A in [QQ, ComplexField(16), GF(5), QQ[sqrt(2)]]:
        print(str(A) + ":")
        print(A['x'](p).factor())
Expected:
    Rational Field:
    (54) * (x + 1/3)^2 * (x^2 - 2)
    Complex Field with 16 bits of precision:
    (54.00) * (x - 1.414) * (x + 0.3333)^2 * (x + 1.414)
    Finite Field of size 5:
    (4) * (x + 2)^2 * (x^2 + 3)
    Number Field in sqrt2 with defining polynomial x^2 - 2:
    (54) * (x - sqrt2) * (x + sqrt2) * (x + 1/3)^2
Got:
    Rational Field:
    (54) * (x + 1/3)^2 * (x^2 - 2)
    Complex Field with 16 bits of precision:
    (54.00) * (x - 1.414) * (x + 0.3333)^2 * (x + 1.414)
    Finite Field of size 5:
    (4) * (x + 2)^2 * (x^2 + 3)
    Number Field in sqrt2 with defining polynomial x^2 - 2 with sqrt2 = 1.414213562373095?:
    (54) * (x - sqrt2) * (x + sqrt2) * (x + 1/3)^2
**********************************************************************
1 item had failures:
   1 of 111 in sage.tests.books.computational-mathematics-with-sagemath.polynomes_doctest
    [110 tests, 1 failure, 0.96 s]
----------------------------------------------------------------------
sage -t --long src/sage/modular/local_comp/smoothchar.py  # 1 doctest failed
sage -t --long src/sage/tests/books/computational-mathematics-with-sagemath/polynomes_doctest.py  # 1 doctest failed
----------------------------------------------------------------------

The first one seems to be taken care of but not the second one.

[git]

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

​9ce74e3

Doctest fix in src/sage/tests/books/computational-mathematics-with-sagemath/polynomes_doctest.py

[git]

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

​76c384e

Fix up doctest after change

[mkoeppe]

Fixed these two.

[git]

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

​c4c0088

Merge tag '8.8.beta5' into t/21161/public/repr_of_numberfields__the_parents__should_indicate_its_embedding_if_there_is_one

[jipilab]

There are only 6 failing doctests. That's not bad!

----------------------------------------------------------------------
sage -t --long src/sage/rings/qqbar.py  # 5 doctests failed
sage -t --long src/sage/rings/number_field/order.py  # 1 doctest failed
----------------------------------------------------------------------

[git]

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

​c1e1514	Merge tag '8.8.beta6' into t/21161/public/repr_of_numberfields__the_parents__should_indicate_its_embedding_if_there_is_one
​d404006	Fix doctests

[jipilab]

Looks good to me. The tests seem to pass on the latest beta and the pyflakes errors are not regressions. I set this to positive review.

[chapoton]

This ticket is probably causing (randomly) infinite loops on several patchbots.

For example, see

​https://patchbot.sagemath.org/log/27408/Ubuntu/18.04/x86_64/4.15.0-52-generic/petitbonum/2019-06-20%2014:50:38?short

EDIT: the problematic doctest is

File "src/sage/structure/parent.pyx", line 1734, in sage.structure.parent.Parent.hom.register_embedding
Failed example:
    K.coerce_embedding()(a)
Exception raised:
...
RuntimeError: maximum recursion depth exceeded while calling a Python object

[jipilab]

Replying to chapoton:

This ticket is probably causing (randomly) infinite loops on several patchbots.

For example, see

​https://patchbot.sagemath.org/log/27408/Ubuntu/18.04/x86_64/4.15.0-52-generic/petitbonum/2019-06-20%2014:50:38?short

EDIT: the problematic doctest is

File "src/sage/structure/parent.pyx", line 1734, in sage.structure.parent.Parent.hom.register_embedding
Failed example:
    K.coerce_embedding()(a)
Exception raised:
...
RuntimeError: maximum recursion depth exceeded while calling a Python object

The source code that changed is

src/sage/rings/number_field/number_field.py

@@ -3087 +3089 @@
         """
-        return "Number Field in %s with defining polynomial %s"%(
-                   self.variable_name(), self.polynomial())
+        result = "Number Field in {} with defining polynomial {}".format(self.variable_name(), self.polynomial())
+        gen = self.gen_embedding()
+        if gen is not None:
+            result += " with {} = {}".format(self.variable_name(), gen)
+        return result

The rest of the diff consist of adapting the doctests. Where could the infinite loop come from? Hmm.

[chapoton]

It appears that using the embedding in the repr is a very bad idea...

Even when this problematic doctest works, calling

K.coerce_embedding()

after this doctest makes sage crash.. And the culprit are really those 2 "innocent" added lines..

[chapoton]

I have made #28036 to revert the changes made here.

[chapoton]

Maybe changing this line would be a way out:

return copy(self._embedding) # It might be overkill to make a copy here