new implicit coercions to QQbar and AA

let python integers and number fields equipped with a complex (resp.
real) embedding coerce into QQbar (AA)

Author: mezzarobba

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/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,
@@ -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 +1177,20 @@ def _coerce_map_from_(self, from_par):
             5.645751311064590?
             sage: AA.has_coerce_map_from(SR)
             False
+
+            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 +1518,15 @@ 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 +1592,28 @@ 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"""