{Polyhedron_base, Polyhedron_base}.base_extend: Handle backend.

src/sage/geometry/polyhedron/base.py

@@ -385,6 +385,8 @@ def base_extend(self, base_ring, backend=None):
         """
         Return a new polyhedron over a larger base ring.
 
+        This method can also be used to change the backend.
+
         INPUT:
 
         - ``base_ring`` -- the new base ring.
@@ -394,7 +396,7 @@ def base_extend(self, base_ring, backend=None):
 
         OUTPUT:
 
-        The same polyhedron, but over a larger base ring.
+        The same polyhedron, but over a larger base ring and possibly with a changed backend.
 
         EXAMPLES::
 
@@ -404,6 +406,15 @@ def base_extend(self, base_ring, backend=None):
             A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray
             sage: P.base_extend(QQ) == P
             True
+
+        TESTS:
+
+        Test that :trac:`22575` is fixed::
+
+            sage: Q = P.base_extend(ZZ, backend='field')
+            sage: Q.backend()
+            'field'
+
         """
         new_parent = self.parent().base_extend(base_ring, backend)
         return new_parent(self)

src/sage/geometry/polyhedron/parent.py

@@ -541,11 +541,21 @@ def base_extend(self, base_ring, backend=None):
             Polyhedra in QQ^3
             sage: Polyhedra(ZZ,3).an_element().base_extend(QQ)
             A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
+
+        TESTS:
+
+        Test that :trac:`22575` is fixed::
+
+            sage: P = Polyhedra(ZZ,3).base_extend(QQ, backend='field')
+            sage: P.backend()
+            'field'
+
         """
-        if self.base_ring().has_coerce_map_from(base_ring):
+        if (self.base_ring().has_coerce_map_from(base_ring)
+            and (backend is None or self.backend() == backend)):
             return self
         elif base_ring.has_coerce_map_from(self.base_ring()):
-            return Polyhedra(base_ring, self.ambient_dim())
+            return Polyhedra(base_ring, self.ambient_dim(), backend=backend)
 
     def _coerce_base_ring(self, other):
         r"""