Trac #23851: Fix memoryleak introduced in #11670

In #11670 the following leak was introduced:
{{{
        sage: u=id(NumberField(x^2-5,'a').absolute_field('b'))
        sage: import gc
        sage: gc.collect() #random
        10
        sage: [id(v) for v in gc.get_objects() if id(v) == u]
}}}

See Nils's explanation in comment:14:ticket:23807 and the surrounding
discussion, where this bug was found.

URL: https://trac.sagemath.org/23851
Reported by: nbruin
Ticket author(s): Nils Bruin, Peter Bruin
Reviewer(s): Sébastien Labbé

src/sage/rings/number_field/number_field.py

@@ -2261,7 +2261,6 @@ def is_relative(self):
         """
         return not self.is_absolute()
 
-
     def quadratic_defect(self, a, p):
         r"""
         Return the valuation of the quadratic defect of `a` at `p`.
@@ -2334,21 +2333,21 @@ def quadratic_defect(self, a, p):
             return v + w
         return Infinity
 
-    @cached_method
     def absolute_field(self, names):
         """
-        Returns self as an absolute extension over QQ.
+        Return ``self`` as an absolute number field.
 
-        OUTPUT:
+        INPUT:
 
+        - ``names`` -- string; name of generator of the absolute field
 
-        -  ``K`` - this number field (since it is already
-           absolute)
+        OUTPUT:
 
+        - ``K`` -- this number field (since it is already absolute)
 
-        Also, ``K.structure()`` returns from_K and to_K,
-        where from_K is an isomorphism from K to self and to_K is an
-        isomorphism from self to K.
+        Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where
+        ``from_K`` is an isomorphism from `K` to ``self`` and ``to_K``
+        is an isomorphism from ``self`` to `K`.
 
         EXAMPLES::
 

src/sage/rings/number_field/number_field_rel.py

@@ -1744,22 +1744,21 @@ def absolute_generator(self):
             self.__abs_gen = self._element_class(self, QQ['x'].gen())
             return self.__abs_gen
 
-    @cached_method
     def absolute_field(self, names):
-        r"""
-        Return an absolute number field `K` that is isomorphic to this
-        field along with a field-theoretic bijection from self to `K`
-        and from `K` to self.
+        """
+        Return ``self`` as an absolute number field.
 
         INPUT:
 
         - ``names`` -- string; name of generator of the absolute field
 
-        OUTPUT: an absolute number field
+        OUTPUT:
+
+        An absolute number field `K` that is isomorphic to this field.
 
         Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where
-        ``from_K`` is an isomorphism from `K` to self and ``to_K`` is
-        an isomorphism from self to `K`.
+        ``from_K`` is an isomorphism from `K` to ``self`` and ``to_K``
+        is an isomorphism from ``self`` to `K`.
 
         EXAMPLES::
 

src/sage/rings/number_field/structure.py

@@ -152,6 +152,15 @@ class NameChange(NumberFieldStructure):
         sage: NameChange(K)
         <sage.rings.number_field.structure.NameChange object at 0x...>
 
+    Check for memory leaks:
+
+        sage: u=id(NumberField(x^2-5,'a').absolute_field('b'))
+        sage: import gc
+        sage: gc.collect() #random
+        10
+        sage: [id(v) for v in gc.get_objects() if id(v) == u]
+        []
+
     """
     def create_structure(self, field):
         r"""

src/sage/rings/polynomial/polynomial_quotient_ring.py

@@ -1386,37 +1386,36 @@ def S_units(self, S, proof=True):
             sage: K.unit_group()
             Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
             sage: K.<a> = QQ['x'].quotient(x^2 + 3)
-            sage: u,o = K.S_units([])[0]; u, o
-            (-1/2*a + 1/2, 6)
+            sage: u,o = K.S_units([])[0]; o
+            6
+            sage: 2*u - 1 in {a, -a}
+            True
             sage: u^6
             1
             sage: u^3
             -1
-            sage: u^2
-            -1/2*a - 1/2
+            sage: 2*u^2 + 1 in {a, -a}
+            True
 
         ::
 
             sage: K.<a> = QuadraticField(-3)
             sage: y = polygen(K)
             sage: L.<b> = K['y'].quotient(y^3 + 5); L
             Univariate Quotient Polynomial Ring in b over Number Field in a with defining polynomial x^2 + 3 with modulus y^3 + 5
-            sage: L.S_units([])
-            [(-1/2*a + 1/2, 6),
-             ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
-             (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
-            sage: L.S_units([K.ideal(1/2*a - 3/2)])
-            [((-1/6*a - 1/2)*b^2 + (1/3*a - 1)*b + 4/3*a, +Infinity),
-             (-1/2*a + 1/2, 6),
-             ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
-             (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
-            sage: L.S_units([K.ideal(2)])
-            [((1/2*a - 1/2)*b^2 + (a + 1)*b + 3, +Infinity),
-             ((1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a + 1/2, +Infinity),
-             ((1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a - 1/2, +Infinity),
-             (-1/2*a + 1/2, 6),
-             ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
-             (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
+            sage: [u for u, o in L.S_units([]) if o is Infinity]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
+             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            sage: [u for u, o in L.S_units([K.ideal(1/2*a - 3/2)]) if o is Infinity]
+            [(-1/6*a - 1/2)*b^2 + (1/3*a - 1)*b + 4/3*a,
+             (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
+             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            sage: [u for u, o in L.S_units([K.ideal(2)]) if o is Infinity]
+            [(1/2*a - 1/2)*b^2 + (a + 1)*b + 3,
+             (1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a + 1/2,
+             (1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a - 1/2,
+             (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
+             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
 
         Note that all the returned values live where we expect them to::
 
@@ -1473,14 +1472,15 @@ def units(self, proof=True):
             sage: K.unit_group()
             Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
             sage: K.<a> = QQ['x'].quotient(x^2 + 3)
-            sage: u = K.units()[0][0]; u
-            -1/2*a + 1/2
+            sage: u = K.units()[0][0]
+            sage: 2*u - 1 in {a, -a}
+            True
             sage: u^6
             1
             sage: u^3
             -1
-            sage: u^2
-            -1/2*a - 1/2
+            sage: 2*u^2 + 1 in {a, -a}
+            True
             sage: K.<a> = QQ['x'].quotient(x^2 + 5)
             sage: K.units(())
             [(-1, 2)]
@@ -1491,17 +1491,16 @@ def units(self, proof=True):
             sage: y = polygen(K)
             sage: L.<b> = K['y'].quotient(y^3 + 5); L
             Univariate Quotient Polynomial Ring in b over Number Field in a with defining polynomial x^2 + 3 with modulus y^3 + 5
-            sage: L.units()
-            [(-1/2*a + 1/2, 6),
-             ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
-             (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
+            sage: [u for u, o in L.units() if o is Infinity]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
+             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
             sage: L.<b> = K.extension(y^3 + 5)
             sage: L.unit_group()
             Unit group with structure C6 x Z x Z of Number Field in b with defining polynomial x^3 + 5 over its base field
             sage: L.unit_group().gens()    # abstract generators
             (u0, u1, u2)
-            sage: L.unit_group().gens_values()
-            [1/2*a + 1/2, (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            sage: L.unit_group().gens_values()[1:]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
 
         Note that all the returned values live where we expect them to::