diff --git a/Convex_hull_3/include/CGAL/Convex_hull_3/dual/halfspace_intersection_interior_point_3.h b/Convex_hull_3/include/CGAL/Convex_hull_3/dual/halfspace_intersection_interior_point_3.h
index afdab1fdee08..ba396c0e2e02 100644
--- a/Convex_hull_3/include/CGAL/Convex_hull_3/dual/halfspace_intersection_interior_point_3.h
+++ b/Convex_hull_3/include/CGAL/Convex_hull_3/dual/halfspace_intersection_interior_point_3.h
@@ -180,7 +180,7 @@ halfspace_intersection_interior_point_3(PlaneIterator begin, PlaneIterator end)
   typedef typename Kernel_traits<typename std::iterator_traits<PlaneIterator>::value_type>::Kernel K;
 
   // choose exact integral type
-  typedef typename internal::Exact_field_selector<void*>::Type ET;
+  typedef typename internal::Exact_field_selector<typename K::FT>::Type ET;
 
   // find a point inside the intersection
   internal::Interior_polyhedron_3<K, ET> interior;
diff --git a/Installation/CHANGES.md b/Installation/CHANGES.md
index 388f336227b9..c52173c61a70 100644
--- a/Installation/CHANGES.md
+++ b/Installation/CHANGES.md
@@ -15,7 +15,7 @@ Release date: October 2023
 [Release 5.6](https://github.com/CGAL/cgal/releases/tag/v5.6)
 -----------
 
-Release date: June 2023
+Release date: July 2023
 
 ### General Changes
 
@@ -186,6 +186,13 @@ Release date: June 2023
     to describe 3D simplicial meshes, and makes the data structure independent
     from the [tetrahedral mesh generation](https://doc.cgal.org/5.6/Manual/packages.html#PkgMesh3) package.
 
+### [Tetrahedral Remeshing](https://doc.cgal.org/5.6/Manual/packages.html#PkgTetrahedralRemeshing)
+-   **Breaking change**: The template parameters of
+    [`CGAL::Tetrahedral_remeshing::Remeshing_vertex_base_3`](https://doc.cgal.org/5.6/Tetrahedral_remeshing/group__PkgTetrahedralRemeshingClasses.html#ga7ef4f8c0c1ed715c34389ea4ee851a92)
+    and
+    [`CGAL::Tetrahedral_remeshing::Remeshing_cell_base_3`](https://doc.cgal.org/5.6/Tetrahedral_remeshing/classCGAL_1_1Tetrahedral__remeshing_1_1Remeshing__cell__base__3.html)
+    have been modified.
+
 ### [3D Mesh Generation](https://doc.cgal.org/5.6/Manual/packages.html#PkgMesh3)
 
 -   Added two new named parameters to the named constructor
diff --git a/Intersections_2/test/Intersections_2/test_intersections_2.cpp b/Intersections_2/test/Intersections_2/test_intersections_2.cpp
index 5c15555404af..dad2d129c15f 100644
--- a/Intersections_2/test/Intersections_2/test_intersections_2.cpp
+++ b/Intersections_2/test/Intersections_2/test_intersections_2.cpp
@@ -924,9 +924,9 @@ int main()
 {
   CGAL::Set_ieee_double_precision pfr;
 
-  Test< CGAL::Simple_cartesian<CGAL::internal::Exact_field_selector<void*>::Type > >().run();
+  Test< CGAL::Simple_cartesian<CGAL::internal::Exact_field_selector<double>::Type > >().run();
   Test< CGAL::Cartesian<double>   >().run();
-  Test< CGAL::Homogeneous<CGAL::internal::Exact_field_selector<void*>::Type > >().run();
+  Test< CGAL::Homogeneous<CGAL::internal::Exact_field_selector<double>::Type > >().run();
   Test< CGAL::Exact_predicates_inexact_constructions_kernel >().run();
   Test< CGAL::Exact_predicates_exact_constructions_kernel >().run();
 }
diff --git a/Number_types/include/CGAL/Exact_integer.h b/Number_types/include/CGAL/Exact_integer.h
index 3ba3fb9816a6..86b13dd3dc63 100644
--- a/Number_types/include/CGAL/Exact_integer.h
+++ b/Number_types/include/CGAL/Exact_integer.h
@@ -40,7 +40,7 @@ typedef unspecified_type Exact_integer;
 
 #else // not DOXYGEN_RUNNING
 
-typedef internal::Exact_ring_selector<int>::Type Exact_integer;
+using Exact_integer = internal::Exact_NT_backend<internal::Default_exact_nt_backend>::Integer;
 
 #endif // not DOXYGEN_RUNNING
 
diff --git a/Number_types/include/CGAL/Exact_rational.h b/Number_types/include/CGAL/Exact_rational.h
index f07c61d42a95..7b53700b5b49 100644
--- a/Number_types/include/CGAL/Exact_rational.h
+++ b/Number_types/include/CGAL/Exact_rational.h
@@ -42,7 +42,7 @@ typedef unspecified_type Exact_rational;
 
 #else // not DOXYGEN_RUNNING
 
-typedef internal::Exact_field_selector<double>::Type Exact_rational;
+using Exact_rational = internal::Exact_NT_backend<internal::Default_exact_nt_backend>::Rational;
 
 #endif
 
diff --git a/Number_types/include/CGAL/Number_types/internal/Exact_type_selector.h b/Number_types/include/CGAL/Number_types/internal/Exact_type_selector.h
index f0d8e961c325..b0f645eb3a6e 100644
--- a/Number_types/include/CGAL/Number_types/internal/Exact_type_selector.h
+++ b/Number_types/include/CGAL/Number_types/internal/Exact_type_selector.h
@@ -156,25 +156,27 @@ constexpr ENT_backend_choice Default_exact_nt_backend = static_cast<ENT_backend_
 #endif
 
 template < typename >
-struct Exact_field_selector
-{
-  using Type = typename Exact_NT_backend<Default_exact_nt_backend>::Rational;
-};
+struct Exact_field_selector;
 
 template < typename >
-struct Exact_ring_selector
-{
-  using Type = typename Exact_NT_backend<Default_exact_nt_backend>::Integer;
-};
-
-template <>
-struct Exact_ring_selector<double>
-{
-  using Type = typename Exact_NT_backend<Default_exact_nt_backend>::Ring_for_float;
+struct Exact_ring_selector;
+
+#define CGAL_EXACT_SELECTORS_SPECS(X) \
+template <> \
+struct Exact_ring_selector<X> \
+{ \
+  using Type = typename Exact_NT_backend<Default_exact_nt_backend>::Ring_for_float; \
+}; \
+\
+template <> \
+struct Exact_field_selector<X> \
+{ \
+  using Type = typename Exact_NT_backend<Default_exact_nt_backend>::Rational; \
 };
 
-template <>
-struct Exact_ring_selector<float> : Exact_ring_selector<double> { };
+CGAL_EXACT_SELECTORS_SPECS(double)
+CGAL_EXACT_SELECTORS_SPECS(float)
+CGAL_EXACT_SELECTORS_SPECS(int)
 
 template <>
 struct Exact_field_selector<MP_Float>
@@ -188,6 +190,10 @@ template <>
 struct Exact_field_selector<Quotient<MP_Float> >
 { typedef Quotient<MP_Float> Type; };
 
+template <>
+struct Exact_ring_selector<Quotient<MP_Float> >
+{ typedef Quotient<MP_Float> Type; };
+
 // And we specialize for the following types :
 #ifdef CGAL_USE_GMP
 template <>
@@ -249,6 +255,10 @@ struct Exact_ring_selector<leda_rational>
 template <>
 struct Exact_field_selector<leda_real>
 { typedef leda_real  Type; };
+
+template <>
+struct Exact_ring_selector<leda_real>
+{ typedef leda_real  Type; };
 #endif
 
 #ifdef CGAL_USE_CORE
@@ -277,6 +287,26 @@ struct Exact_field_selector<Exact_NT_backend<BOOST_BACKEND>::Rational>
 template <>
 struct Exact_ring_selector<Exact_NT_backend<BOOST_BACKEND>::Rational>
 { typedef Exact_NT_backend<BOOST_BACKEND>::Rational  Type; };
+
+
+#ifdef CGAL_USE_GMP
+template <>
+struct Exact_field_selector<Exact_NT_backend<BOOST_GMP_BACKEND>::Integer>
+{ typedef Exact_NT_backend<BOOST_GMP_BACKEND>::Rational  Type; };
+
+template <>
+struct Exact_ring_selector<Exact_NT_backend<BOOST_GMP_BACKEND>::Integer>
+{ typedef Exact_NT_backend<BOOST_GMP_BACKEND>::Integer  Type; };
+
+template <>
+struct Exact_field_selector<Exact_NT_backend<BOOST_GMP_BACKEND>::Rational>
+{ typedef Exact_NT_backend<BOOST_GMP_BACKEND>::Rational  Type; };
+
+template <>
+struct Exact_ring_selector<Exact_NT_backend<BOOST_GMP_BACKEND>::Rational>
+{ typedef Exact_NT_backend<BOOST_GMP_BACKEND>::Rational  Type; };
+#endif
+
 #endif
 
 #ifndef CGAL_NO_DEPRECATED_CODE
@@ -287,4 +317,6 @@ struct Exact_type_selector : Exact_field_selector< ET > {};
 
 } } // namespace CGAL::internal
 
+#undef CGAL_EXACT_SELECTORS_SPECS
+
 #endif // CGAL_INTERNAL_EXACT_TYPE_SELECTOR_H