Trac #24116: Fix Cython warnings in finite_rings

Fix all warnings "Compatible but non-identical C method ... not
redeclared" and "Overriding cdef method with def method"

This is part of #23600

URL: https://trac.sagemath.org/24116
Reported by: jdemeyer
Ticket author(s): Jeroen Demeyer
Reviewer(s): Travis Scrimshaw

src/sage/rings/finite_rings/element_base.pxd

@@ -1,7 +1,8 @@
 from sage.structure.element cimport CommutativeRingElement
 
 cdef class FiniteRingElement(CommutativeRingElement):
-    pass
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
 
 cdef class FinitePolyExtElement(FiniteRingElement):
     pass

src/sage/rings/finite_rings/element_base.pyx

@@ -28,6 +28,7 @@ def is_FiniteFieldElement(x):
     from sage.rings.finite_rings.finite_field_base import is_FiniteField
     return isinstance(x, Element) and is_FiniteField(x.parent())
 
+
 cdef class FiniteRingElement(CommutativeRingElement):
     def _nth_root_common(self, n, all, algorithm, cunningham):
         """
@@ -99,6 +100,15 @@ cdef class FiniteRingElement(CommutativeRingElement):
         else:
             raise ValueError("unknown algorithm")
 
+    cpdef _add_(self, other):
+        """Abstract addition method"""
+        raise NotImplementedError
+
+    cpdef _mul_(self, other):
+        """Abstract multiplication method"""
+        raise NotImplementedError
+
+
 cdef class FinitePolyExtElement(FiniteRingElement):
     """
     Elements represented as polynomials modulo a given ideal.
@@ -795,4 +805,3 @@ cdef class FinitePolyExtElement(FiniteRingElement):
             b^11 + b^10 + b^9 + b^7 + b^5 + b^4 + b^2 + b
         """
         return self.pth_power(-k)
-

src/sage/rings/finite_rings/element_givaro.pyx

@@ -1130,7 +1130,6 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
                                               (<FiniteField_givaroElement>right).element)
         return make_FiniteField_givaroElement(self._cache,r)
 
-
     cpdef _div_(self, right):
         """
         Divide two elements

src/sage/rings/finite_rings/finite_field_base.pyx

@@ -1073,7 +1073,7 @@ cdef class FiniteField(Field):
                       and hasattr(self, '_prefix') and hasattr(R, '_prefix')):
                     return R.hom((self.gen() ** ((self.order() - 1)//(R.order() - 1)),))
 
-    def _convert_map_from_(self, R):
+    cpdef _convert_map_from_(self, R):
         """
         Conversion from p-adic fields.