moving "is_integrally_closed" to the category setup.

src/doc/en/thematic_tutorials/coercion_and_categories.rst

@@ -132,7 +132,6 @@ This base class provides a lot more methods than a general parent::
      'integral_closure',
      'is_commutative',
      'is_field',
-     'is_integrally_closed',
      'krull_dimension',
      'localization',
      'ngens',

src/sage/categories/fields.py

@@ -206,11 +206,12 @@ def is_field(self, proof=True):
             """
             return True
 
-        def is_integrally_closed(self):
+        def is_integrally_closed(self) -> bool:
             r"""
-            Return ``True``, as per :meth:`IntegralDomain.is_integrally_closed`:
-            for every field `F`, `F` is its own field of fractions,
-            hence every element of `F` is integral over `F`.
+            Return whether ``self`` is integrally closed.
+
+            For every field `F`, `F` is its own field of fractions.
+            Therefore every element of `F` is integral over `F`.
 
             EXAMPLES::
 
@@ -222,6 +223,8 @@ def is_integrally_closed(self):
                 Finite Field of size 5
                 sage: Z5.is_integrally_closed()
                 True
+                sage: Frac(ZZ['x,y']).is_integrally_closed()
+                True
             """
             return True
 

src/sage/categories/integral_domains.py

@@ -1,6 +1,27 @@
 # sage_setup: distribution = sagemath-categories
 r"""
 Integral domains
+
+TEST:
+
+A few tests for the method ``is_integrally_closed``::
+
+    sage: ZZ.is_integrally_closed()
+    True
+    sage: QQ.is_integrally_closed()
+    True
+    sage: QQbar.is_integrally_closed()                                          # needs sage.rings.number_field
+    True
+    sage: GF(5).is_integrally_closed()
+    True
+    sage: Z5 = Integers(5); Z5
+    Ring of integers modulo 5
+    sage: Z5.is_integrally_closed()
+    Traceback (most recent call last):
+    ...
+    NotImplementedError
+
+Note that this raises a :exc:`NotImplementedError` if the answer is not known.
 """
 # ****************************************************************************
 #  Copyright (C) 2008 Teresa Gomez-Diaz (CNRS) <[email protected]>

src/sage/categories/rings.py

@@ -423,6 +423,29 @@ def is_integral_domain(self, proof=True) -> bool:
 
             return False
 
+        def is_integrally_closed(self) -> bool:
+            r"""
+            Return whether this ring is integrally closed.
+
+            This is the default implementation that
+            raises a :exc:`NotImplementedError`.
+
+            EXAMPLES::
+
+                sage: x = polygen(ZZ, 'x')
+                sage: K.<a> = NumberField(x^2 + 189*x + 394)
+                sage: R = K.order(2*a)
+                sage: R.is_integrally_closed()
+                False
+                sage: R
+                Order of conductor 2 generated by 2*a in Number Field in a with defining polynomial x^2 + 189*x + 394
+                sage: S = K.maximal_order(); S
+                Maximal Order generated by a in Number Field in a with defining polynomial x^2 + 189*x + 394
+                sage: S.is_integrally_closed()
+                True
+            """
+            raise NotImplementedError
+
         def is_noetherian(self):
             """
             Return ``True`` if this ring is Noetherian.

src/sage/rings/integer_ring.pyx

@@ -1142,22 +1142,6 @@ cdef class IntegerRing_class(CommutativeRing):
         """
         return 1
 
-    def is_integrally_closed(self):
-        """
-        Return that the integer ring is, in fact, integrally closed.
-
-        .. NOTE::
-
-            This should rather be inherited from the category
-            of ``DedekindDomains``.
-
-        EXAMPLES::
-
-            sage: ZZ.is_integrally_closed()
-            True
-        """
-        return True
-
     def completion(self, p, prec, extras = {}):
         r"""
         Return the metric completion of the integers at the prime `p`.

src/sage/rings/number_field/order.py

@@ -631,9 +631,11 @@ def is_noetherian(self):
         """
         return True
 
-    def is_integrally_closed(self):
+    def is_integrally_closed(self) -> bool:
         r"""
-        Return ``True`` if this ring is integrally closed, i.e., is equal
+        Return whether this ring is integrally closed.
+
+        This is true if and only if it is equal
         to the maximal order.
 
         EXAMPLES::

src/sage/rings/ring.pyx

@@ -1018,37 +1018,6 @@ cdef class IntegralDomain(CommutativeRing):
         CommutativeRing.__init__(self, base_ring, names=names, normalize=normalize,
                                  category=category)
 
-    def is_integrally_closed(self):
-        r"""
-        Return ``True`` if this ring is integrally closed in its field of
-        fractions; otherwise return ``False``.
-
-        When no algorithm is implemented for this, then this
-        function raises a :exc:`NotImplementedError`.
-
-        Note that ``is_integrally_closed`` has a naive implementation
-        in fields. For every field `F`, `F` is its own field of fractions,
-        hence every element of `F` is integral over `F`.
-
-        EXAMPLES::
-
-            sage: ZZ.is_integrally_closed()
-            True
-            sage: QQ.is_integrally_closed()
-            True
-            sage: QQbar.is_integrally_closed()                                          # needs sage.rings.number_field
-            True
-            sage: GF(5).is_integrally_closed()
-            True
-            sage: Z5 = Integers(5); Z5
-            Ring of integers modulo 5
-            sage: Z5.is_integrally_closed()
-            Traceback (most recent call last):
-            ...
-            AttributeError: 'IntegerModRing_generic_with_category' object has no attribute 'is_integrally_closed'...
-        """
-        raise NotImplementedError
-
     def is_field(self, proof=True):
         r"""
         Return ``True`` if this ring is a field.
@@ -1228,18 +1197,6 @@ cdef class Field(CommutativeRing):
         """
         return True
 
-    def is_integrally_closed(self):
-        """
-        Return ``True`` since fields are trivially integrally closed in
-        their fraction field (since they are their own fraction field).
-
-        EXAMPLES::
-
-            sage: Frac(ZZ['x,y']).is_integrally_closed()
-            True
-        """
-        return True
-
     def krull_dimension(self):
         """
         Return the Krull dimension of this field, which is 0.