Trac #31628: fix/improve conversions to QQbar and AA

* fix conversion from ℚ[i] to `QQbar` (honor complex embeddings)
* add implicit coercions from python ints and from number fields with
compatible embeddings

URL: https://trac.sagemath.org/31628
Reported by: mmezzarobba
Ticket author(s): Marc Mezzarobba, Vincent Delecroix
Reviewer(s): Vincent Delecroix, Marc Mezzarobba

Author: Release Manager

src/sage/modular/dirichlet.py

@@ -1590,10 +1590,10 @@ def kloosterman_sum(self, a=1, b=0):
 
             sage: G = DirichletGroup(12, QQbar)
             sage: e = G.gens()[0]
+            sage: e.kloosterman_sum(5, 4)
+            0.?e-17 - 4.000000000000000?*I
             sage: e.kloosterman_sum(5,11)
-            Traceback (most recent call last):
-            ...
-            NotImplementedError: Kloosterman sums not implemented over this ring
+            0
         """
         G = self.parent()
         zo = G.zeta_order()

src/sage/rings/number_field/number_field_element.pyx

@@ -2789,6 +2789,10 @@ cdef class NumberFieldElement(FieldElement):
             sage: E = C.algebraic_closure()
             sage: E(a)
             -4.949886207424724? - 0.2195628712241434?*I
+
+            sage: NF.<sqrt2> = QuadraticField(2)
+            sage: AA(sqrt2)
+            1.414213562373095?
         """
         if self.is_rational():
             return parent(self._rational_())

src/sage/rings/number_field/number_field_element_quadratic.pyx

@@ -1706,29 +1706,6 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
             mpq_canonicalize(res.value)
             return res
 
-    def _algebraic_(self, parent):
-        r"""
-        Convert this element to an algebraic number, if possible.
-
-        EXAMPLES::
-
-            sage: NF.<i> = QuadraticField(-1)
-            sage: QQbar(1+i)
-            I + 1
-            sage: NF.<sqrt3> = QuadraticField(2)
-            sage: AA(sqrt3)
-            1.414213562373095?
-        """
-        import sage.rings.qqbar as qqbar
-        if (parent is qqbar.QQbar
-                and list(self._parent.polynomial()) == [1, 0, 1]):
-            # AlgebraicNumber.__init__ does a better job than
-            # NumberFieldElement._algebraic_ in this case, but
-            # QQbar._element_constructor_ calls the latter first.
-            return qqbar.AlgebraicNumber(self)
-        else:
-            return NumberFieldElement._algebraic_(self, parent)
-
     cpdef bint is_one(self):
         r"""
         Check whether this number field element is `1`.
@@ -2417,6 +2394,34 @@ cdef class NumberFieldElement_gaussian(NumberFieldElement_quadratic):
         from sage.symbolic.constants import I
         return self[1]*(I if self.standard_embedding else -I) + self[0]
 
+    def _algebraic_(self, parent):
+        r"""
+        Convert this element to an algebraic number, if possible.
+
+        EXAMPLES::
+
+            sage: NF.<i> = QuadraticField(-1)
+            sage: QQbar(i+2)
+            I + 2
+            sage: K.<ii> = QuadraticField(-1, embedding=CC(0,-1))
+            sage: QQbar(ii+2)
+            -I + 2
+            sage: AA(i)
+            Traceback (most recent call last):
+            ...
+            ValueError: unable to convert i to an element of Algebraic Real Field
+        """
+        import sage.rings.qqbar as qqbar
+        if parent is qqbar.QQbar:
+            # AlgebraicNumber.__init__ does a better job than
+            # NumberFieldElement._algebraic_ in this case, but
+            # QQbar._element_constructor_ calls the latter first.
+            return qqbar.AlgebraicNumber(self)
+        elif parent is qqbar.AA and self[1].is_zero():
+            return qqbar.AlgebraicReal(self)
+        else:
+            raise ValueError(f"unable to convert {self!r} to an element of {parent!r}")
+
     cpdef real_part(self):
         r"""
         Real part.

src/sage/rings/qqbar.py

@@ -115,10 +115,6 @@
     3.146264369941973?
     sage: QQbar(I)
     I
-    sage: AA(I)
-    Traceback (most recent call last):
-    ...
-    ValueError: Cannot coerce algebraic number with non-zero imaginary part to algebraic real
     sage: QQbar(I * golden_ratio)
     1.618033988749895?*I
     sage: AA(golden_ratio)^2 - AA(golden_ratio)
@@ -150,24 +146,30 @@
     sage: QQbar(-1)^(1/3)
     0.500000000000000? + 0.866025403784439?*I
 
-We can explicitly coerce from `\QQ[I]`. (Technically, this is not quite
-kosher, since `\QQ[I]` does not come with an embedding; we do not know
-whether the field generator is supposed to map to `+I` or `-I`. We assume
-that for any quadratic field with polynomial `x^2+1`, the generator maps
-to `+I`.)::
-
-    sage: K.<im> = QQ[I]
-    sage: pythag = QQbar(3/5 + 4*im/5); pythag
-    4/5*I + 3/5
-    sage: pythag.abs() == 1
-    True
+However, implicit coercion from `\QQ[I]` is only allowed when it is equipped
+with a complex embedding::
 
-However, implicit coercion from `\QQ[I]` is not allowed::
+    sage: i.parent()
+    Number Field in I with defining polynomial x^2 + 1 with I = 1*I
+    sage: QQbar(1) + i
+    I + 1
 
+    sage: K.<im> = QuadraticField(-1, embedding=None)
     sage: QQbar(1) + im
     Traceback (most recent call last):
     ...
-    TypeError: unsupported operand parent(s) for +: 'Algebraic Field' and 'Number Field in I with defining polynomial x^2 + 1 with I = 1*I'
+    TypeError: unsupported operand parent(s) for +: 'Algebraic Field' and
+    'Number Field in im with defining polynomial x^2 + 1'
+
+However, we can explicitly coerce from the abstract number field `\QQ[I]`.
+(Technically, this is not quite kosher, since we do not know whether the field
+generator is supposed to map to `+I` or `-I`. We assume that for any quadratic
+field with polynomial `x^2+1`, the generator maps to `+I`.)::
+
+    sage: pythag = QQbar(3/5 + 4*im/5); pythag
+    4/5*I + 3/5
+    sage: pythag.abs() == 1
+    True
 
 We can implicitly coerce from algebraic reals to algebraic numbers::
 
@@ -553,8 +555,10 @@
 import operator
 
 import sage.rings.ring
+import sage.rings.number_field.number_field_base
 from sage.misc.fast_methods import Singleton
 from sage.misc.cachefunc import cached_method
+from sage.structure.coerce import parent_is_numerical, parent_is_real_numerical
 from sage.structure.sage_object import SageObject
 from sage.structure.richcmp import (richcmp, richcmp_method,
                                     rich_to_bool, richcmp_not_equal,
@@ -569,7 +573,7 @@
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
 from sage.rings.number_field.number_field import NumberField, GaussianField, CyclotomicField
-from sage.rings.number_field.number_field_element_quadratic import NumberFieldElement_quadratic
+from sage.rings.number_field.number_field_element_quadratic import NumberFieldElement_quadratic, NumberFieldElement_gaussian
 from sage.arith.all import factor
 from . import infinity
 from sage.categories.action import Action
@@ -1063,9 +1067,17 @@ def __init__(self):
 
             sage: QQbar.category() # indirect doctest
             Category of infinite fields
+
+        Coercions::
+
+            sage: AA.has_coerce_map_from(ZZ)
+            True
+            sage: AA.has_coerce_map_from(int)
+            True
         """
         from sage.categories.fields import Fields
         AlgebraicField_common.__init__(self, self, ('x',), normalize=False, category=Fields().Infinite())
+        self._populate_coercion_lists_([ZZ, QQ])
 
     def _element_constructor_(self, x):
         r"""
@@ -1157,8 +1169,6 @@ def _coerce_map_from_(self, from_par):
 
         TESTS::
 
-            sage: AA.has_coerce_map_from(ZZ) # indirect doctest
-            True
             sage: K.<a> = QuadraticField(7, embedding=AA(7).sqrt()); AA.has_coerce_map_from(K)
             True
             sage: a in AA
@@ -1167,9 +1177,23 @@ def _coerce_map_from_(self, from_par):
             5.645751311064590?
             sage: AA.has_coerce_map_from(SR)
             False
+
+            sage: K = NumberField(x^3 - 2, 'a', embedding=2.**(1/3))
+            sage: AA.has_coerce_map_from(K)
+            True
+            sage: K.<s> = QuadraticField(3, embedding=-2.)
+            sage: s + AA(1)
+            -0.732050807568878?
+            sage: K.<s> = QuadraticField(3, embedding=2.)
+            sage: s + AA(1)
+            2.732050807568878?
+            sage: K.<s> = QuadraticField(-5)
+            sage: AA.has_coerce_map_from(K)
+            False
         """
-        return (from_par is ZZ or from_par is QQ
-                or from_par is AA)
+        if isinstance(from_par, sage.rings.number_field.number_field_base.NumberField):
+            emb = from_par.coerce_embedding()
+            return emb is not None and parent_is_real_numerical(emb.codomain())
 
     def completion(self, p, prec, extras={}):
         r"""
@@ -1497,9 +1521,15 @@ def __init__(self):
 
             sage: QQbar._repr_option('element_is_atomic')
             False
+
+            sage: QQbar.has_coerce_map_from(ZZ)
+            True
+            sage: QQbar.has_coerce_map_from(int)
+            True
         """
         from sage.categories.fields import Fields
         AlgebraicField_common.__init__(self, AA, ('I',), normalize=False, category=Fields().Infinite())
+        self._populate_coercion_lists_([ZZ, QQ])
 
     def _element_constructor_(self, x):
         """
@@ -1565,17 +1595,28 @@ def _coerce_map_from_(self, from_par):
 
         TESTS::
 
-            sage: QQbar.has_coerce_map_from(ZZ) # indirect doctest
-            True
             sage: QQbar.has_coerce_map_from(AA)
             True
             sage: QQbar.has_coerce_map_from(CC)
             False
             sage: QQbar.has_coerce_map_from(SR)
             False
+
+            sage: i + QQbar(2)
+            I + 2
+            sage: K.<ii> = QuadraticField(-1, embedding=ComplexField(13)(0,-1))
+            sage: ii + QQbar(2)
+            -I + 2
+
+            sage: L.<a> = QuadraticField(-1, embedding=Zp(5).teichmuller(2))
+            sage: QQbar.has_coerce_map_from(L)
+            False
         """
-        return (from_par is ZZ or from_par is QQ
-                or from_par is AA or from_par is QQbar)
+        if from_par is AA:
+            return True
+        if isinstance(from_par, sage.rings.number_field.number_field_base.NumberField):
+            emb = from_par.coerce_embedding()
+            return emb is not None and parent_is_numerical(emb.codomain())
 
     def completion(self, p, prec, extras={}):
         r"""
@@ -3453,10 +3494,11 @@ def __init__(self, parent, x):
             self._descr = ANRational(x)
         elif isinstance(x, ANDescr):
             self._descr = x
-        elif parent is QQbar and \
-                 isinstance(x, NumberFieldElement_quadratic) and \
-                 list(x.parent().polynomial()) == [1, 0, 1]:
-            self._descr = ANExtensionElement(QQbar_I_generator, QQbar_I_nf(x.list()))
+        elif parent is QQbar and isinstance(x, NumberFieldElement_gaussian):
+            if x.parent()._standard_embedding:
+                self._descr = ANExtensionElement(QQbar_I_generator, QQbar_I_nf(x.list()))
+            else:
+                self._descr = ANExtensionElement(QQbar_I_generator, QQbar_I_nf([x[0], -x[1]]))
         else:
             raise TypeError("Illegal initializer for algebraic number")
 

src/sage/symbolic/expression.pyx

@@ -3150,13 +3150,6 @@ cdef class Expression(CommutativeRingElement):
             sage: bool(f(x) - f(x) == 0)
             True
 
-        Check that we catch exceptions from Pynac (:trac:`19904`)::
-
-            sage: bool(SR(QQbar(I)) == I)
-            Traceback (most recent call last):
-            ...
-            TypeError: unsupported operand parent(s)...
-
         Check that :trac:`24658` is fixed::
 
             sage: val = pi - 2286635172367940241408/1029347477390786609545*sqrt(2)