moving "is_prime_field" to the category of rings

Author: fchapoton

src/doc/en/thematic_tutorials/coercion_and_categories.rst

@@ -134,7 +134,6 @@ This base class provides a lot more methods than a general parent::
      'is_commutative',
      'is_field',
      'is_integrally_closed',
-     'is_prime_field',
      'krull_dimension',
      'localization',
      'ngens',

src/sage/categories/rings.py

@@ -434,6 +434,30 @@ def is_noetherian(self):
             """
             return False
 
+        def is_prime_field(self):
+            r"""
+            Return ``True`` if this ring is one of the prime fields `\QQ` or
+            `\GF{p}`.
+
+            EXAMPLES::
+
+                sage: QQ.is_prime_field()
+                True
+                sage: GF(3).is_prime_field()
+                True
+                sage: GF(9, 'a').is_prime_field()                                           # needs sage.rings.finite_rings
+                False
+                sage: ZZ.is_prime_field()
+                False
+                sage: QQ['x'].is_prime_field()
+                False
+                sage: Qp(19).is_prime_field()                                               # needs sage.rings.padics
+                False
+            """
+            # the case of QQ is handled by QQ itself
+            from sage.categories.finite_fields import FiniteFields
+            return self in FiniteFields() and self.degree() == 1
+
         def is_zero(self) -> bool:
             """
             Return ``True`` if this is the zero ring.

src/sage/rings/ring.pyx

@@ -652,28 +652,6 @@ cdef class Ring(ParentWithGens):
         """
         return True
 
-    def is_prime_field(self):
-        r"""
-        Return ``True`` if this ring is one of the prime fields `\QQ` or
-        `\GF{p}`.
-
-        EXAMPLES::
-
-            sage: QQ.is_prime_field()
-            True
-            sage: GF(3).is_prime_field()
-            True
-            sage: GF(9, 'a').is_prime_field()                                           # needs sage.rings.finite_rings
-            False
-            sage: ZZ.is_prime_field()
-            False
-            sage: QQ['x'].is_prime_field()
-            False
-            sage: Qp(19).is_prime_field()                                               # needs sage.rings.padics
-            False
-        """
-        return False
-
     def order(self):
         """
         The number of elements of ``self``.