Trac #25377: add 'same_field' option to number_field_elements_from_al…

…gebraics()

src/sage/rings/qqbar.py

@@ -1924,7 +1924,7 @@ def conjugate_shrink(v):
         return v.real()
     return v
 
-def number_field_elements_from_algebraics(numbers, minimal=False):
+def number_field_elements_from_algebraics(numbers, minimal=False, same_field=False):
     r"""
     Given a sequence of elements of either ``AA`` or ``QQbar``
     (or a mixture), computes a number field containing all of these
@@ -1935,6 +1935,13 @@ def number_field_elements_from_algebraics(numbers, minimal=False):
     This may not return the smallest such number field, unless
     ``minimal=True`` is specified.
 
+    If ``same_field=True`` is specified, and all of the elements are
+    from the same field (either ``AA`` or ``QQbar``), the generated
+    homomorphism will map back to that field.  Otherwise, if all specified
+    elements are real, the homomorphism might map back to ``AA``
+    (and will, if ``minimal=True`` is specified), even if the
+    elements were in ``QQbar``.
+
     Also, a single number can be passed, rather than a sequence; and
     any values which are not elements of ``AA`` or ``QQbar``
     will automatically be coerced to ``QQbar``
@@ -1960,6 +1967,8 @@ def number_field_elements_from_algebraics(numbers, minimal=False):
         1.414213562373095?
         sage: rt3 = AA(sqrt(3)); rt3
         1.732050807568878?
+        sage: rt3a = QQbar(sqrt(3)); rt3a
+        1.732050807568878?
         sage: qqI = QQbar.zeta(4); qqI
         I
         sage: z3 = QQbar.zeta(3); z3
@@ -1981,6 +1990,22 @@ def number_field_elements_from_algebraics(numbers, minimal=False):
             To:   Algebraic Real Field
             Defn: a |--> 0.5176380902050415?)
 
+    ``rt3a`` is a real number in ``QQbar``.  Ordinarily, we'd get a homomorphism
+    to ``AA`` (because all elements are real), but if we specify ``same_field=True``,
+    we'll get a homomorphism back to ``QQbar``::
+
+        sage: number_field_elements_from_algebraics(rt3a)
+        (Number Field in a with defining polynomial y^2 - 3, a, Ring morphism:
+            From: Number Field in a with defining polynomial y^2 - 3
+            To:   Algebraic Real Field
+            Defn: a |--> 1.732050807568878?)
+
+        sage: number_field_elements_from_algebraics(rt3a, same_field=True)
+        (Number Field in a with defining polynomial y^2 - 3, a, Ring morphism:
+            From: Number Field in a with defining polynomial y^2 - 3
+            To:   Algebraic Field
+            Defn: a |--> 1.732050807568878?)
+
     We've created ``rt2b`` in such a way that \sage doesn't initially know
     that it's in a degree-2 extension of `\QQ`::
 
@@ -2059,9 +2084,12 @@ def number_field_elements_from_algebraics(numbers, minimal=False):
 
     Note that for the first example, where \sage doesn't realize that
     the number is real, we get a homomorphism to ``QQbar``; but with
-    ``minimal=True``, we get a homomorphism to ``AA``. Also note
-    that the exact answer depends on a Pari function that gives
-    different answers for 32-bit and 64-bit machines::
+    ``minimal=True``, we get a homomorphism to ``AA``.  If we specify
+    both ``minimal=True`` and ``same_field=True``, we get a second
+    degree extension (minimal) that maps back to ``QQbar``.
+
+    Also note that the exact answer depends on a Pari function that
+    gives different answers for 32-bit and 64-bit machines::
 
         sage: number_field_elements_from_algebraics(rt2c)
         (Number Field in a with defining polynomial y^4 + 2*y^2 + 4, 1/2*a^3, Ring morphism:
@@ -2073,6 +2101,11 @@ def number_field_elements_from_algebraics(numbers, minimal=False):
             From: Number Field in a with defining polynomial y^2 - 2
             To:   Algebraic Real Field
             Defn: a |--> 1.414213562373095?)
+        sage: number_field_elements_from_algebraics(rt2c, minimal=True, same_field=True)
+        (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 Field
+            Defn: a |--> 1.414213562373095?)
 
     """
     gen = qq_generator
@@ -2085,6 +2118,11 @@ def number_field_elements_from_algebraics(numbers, minimal=False):
         numbers = [numbers]
         single_number = True
 
+    if any([isinstance(_, AlgebraicNumber) for _ in numbers]):
+        algebraic_field = QQbar
+    else:
+        algebraic_field = AA
+
     def mk_algebraic(x):
         if isinstance(x, AlgebraicNumber_base):
             return x
@@ -2104,7 +2142,10 @@ def mk_algebraic(x):
     if single_number:
         nums = nums[0]
 
-    hom = fld.hom([gen.root_as_algebraic()])
+    if same_field:
+        hom = fld.hom([gen.root_as_algebraic()], codomain=algebraic_field)
+    else:
+        hom = fld.hom([gen.root_as_algebraic()])
 
     return (fld, nums, hom)