gh-39394: move methods into category of fields
    
moving 3 methods from the auld `Field` class to the category of Fields.

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #39394
Reported by: Frédéric Chapoton
Reviewer(s): Vincent Macri

Author: Release Manager

src/doc/en/thematic_tutorials/coercion_and_categories.rst

@@ -123,18 +123,15 @@ This base class provides a lot more methods than a general parent::
      'algebraic_closure',
      'an_embedding',
      'base_extend',
-     'divides',
      'epsilon',
      'extension',
      'fraction_field',
      'gen',
      'gens',
-     'integral_closure',
      'is_field',
      'ngens',
      'one',
      'order',
-     'prime_subfield',
      'random_element',
      'zero',
      'zeta',

src/sage/categories/fields.py

@@ -240,6 +240,61 @@ def is_integrally_closed(self) -> bool:
             """
             return True
 
+        def integral_closure(self):
+            """
+            Return this field, since fields are integrally closed in their
+            fraction field.
+
+            EXAMPLES::
+
+                sage: QQ.integral_closure()
+                Rational Field
+                sage: Frac(ZZ['x,y']).integral_closure()
+                Fraction Field of Multivariate Polynomial Ring in x, y
+                over Integer Ring
+            """
+            return self
+
+        def prime_subfield(self):
+            """
+            Return the prime subfield of ``self``.
+
+            EXAMPLES::
+
+                sage: k = GF(9, 'a')                                                        # needs sage.rings.finite_rings
+                sage: k.prime_subfield()                                                    # needs sage.rings.finite_rings
+                Finite Field of size 3
+            """
+            if self.characteristic() == 0:
+                import sage.rings.rational_field
+                return sage.rings.rational_field.RationalField()
+
+            from sage.rings.finite_rings.finite_field_constructor import GF
+            return GF(self.characteristic())
+
+        def divides(self, x, y, coerce=True):
+            """
+            Return ``True`` if ``x`` divides ``y`` in this field.
+
+            This is usually ``True`` in a field!.
+
+            If ``coerce`` is ``True`` (the default), first coerce
+            ``x`` and ``y`` into ``self``.
+
+            EXAMPLES::
+
+                sage: QQ.divides(2, 3/4)
+                True
+                sage: QQ.divides(0, 5)
+                False
+            """
+            if coerce:
+                x = self(x)
+                y = self(y)
+            if x.is_zero():
+                return y.is_zero()
+            return True
+
         def _gcd_univariate_polynomial(self, a, b):
             """
             Return the gcd of ``a`` and ``b`` as a monic polynomial.

src/sage/rings/ring.pyx

@@ -979,40 +979,6 @@ cdef class Field(CommutativeRing):
         """
         return self
 
-    def divides(self, x, y, coerce=True):
-        """
-        Return ``True`` if ``x`` divides ``y`` in this field (usually ``True``
-        in a field!).  If ``coerce`` is ``True`` (the default), first coerce
-        ``x`` and ``y`` into ``self``.
-
-        EXAMPLES::
-
-            sage: QQ.divides(2, 3/4)
-            True
-            sage: QQ.divides(0, 5)
-            False
-        """
-        if coerce:
-            x = self(x)
-            y = self(y)
-        if x.is_zero():
-            return y.is_zero()
-        return True
-
-    def integral_closure(self):
-        """
-        Return this field, since fields are integrally closed in their
-        fraction field.
-
-        EXAMPLES::
-
-            sage: QQ.integral_closure()
-            Rational Field
-            sage: Frac(ZZ['x,y']).integral_closure()
-            Fraction Field of Multivariate Polynomial Ring in x, y over Integer Ring
-        """
-        return self
-
     def is_field(self, proof=True):
         """
         Return ``True`` since this is a field.
@@ -1024,23 +990,6 @@ cdef class Field(CommutativeRing):
         """
         return True
 
-    def prime_subfield(self):
-        """
-        Return the prime subfield of ``self``.
-
-        EXAMPLES::
-
-            sage: k = GF(9, 'a')                                                        # needs sage.rings.finite_rings
-            sage: k.prime_subfield()                                                    # needs sage.rings.finite_rings
-            Finite Field of size 3
-        """
-        if self.characteristic() == 0:
-            import sage.rings.rational_field
-            return sage.rings.rational_field.RationalField()
-        else:
-            from sage.rings.finite_rings.finite_field_constructor import GF
-            return GF(self.characteristic())
-
     def algebraic_closure(self):
         """
         Return the algebraic closure of ``self``.