Trac #33568: Add method is_integral_domain for polynomial quotient …

…rings

We add a method `is_integral_domain`
to the `PolynomialQuotientRing` class.

URL: https://trac.sagemath.org/33568
Reported by: gh-schmittj
Ticket author(s): Johannes Schmitt
Reviewer(s): Vincent Delecroix

build/pkgs/configure/checksums.ini

@@ -1,4 +1,4 @@
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+sha1=b96c0334857e7afceccdb63b4e9ccb0c2560961f
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build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-f43e375fb59b4b297a3b2b4c3da31f0d4d34adb3
+d7a2f1f4880be729ef4d26ceaf3f1b8af941feb8

src/sage/rings/polynomial/polynomial_quotient_ring.py

@@ -1059,6 +1059,87 @@ def is_field(self, proof = True):
             self._refine_category_(Fields())
         return ret
 
+    def is_integral_domain(self, proof = True):
+        """
+        Return whether or not this quotient ring is an integral domain.
+
+        EXAMPLES::
+
+            sage: R.<z> = PolynomialRing(ZZ)
+            sage: S = R.quotient(z^2 - z)
+            sage: S.is_integral_domain()
+            False
+            sage: T = R.quotient(z^2 + 1)
+            sage: T.is_integral_domain()
+            True
+            sage: U = R.quotient(-1)
+            sage: U.is_integral_domain()
+            False
+            sage: R2.<y> = PolynomialRing(R)
+            sage: S2 = R2.quotient(z^2 - y^3)
+            sage: S2.is_integral_domain()
+            True
+            sage: S3 = R2.quotient(z^2 - 2*y*z + y^2)
+            sage: S3.is_integral_domain()
+            False
+
+            sage: R.<z> = PolynomialRing(ZZ.quotient(4))
+            sage: S = R.quotient(z-1)
+            sage: S.is_integral_domain()
+            False
+
+        TESTS:
+
+        Here is an example of a quotient ring which is not an integral
+        domain, even though the base ring is integral and the modulus is
+        irreducible::
+
+            sage: B = ZZ.extension(x^2 - 5, 'a')
+            sage: R.<y> = PolynomialRing(B)
+            sage: S = R.quotient(y^2 - y - 1)
+            sage: S.is_integral_domain()
+            Traceback (most recent call last):
+            ...
+            NotImplementedError
+            sage: S.is_integral_domain(proof = False)
+            False
+
+        The reason that the modulus y^2 - y -1 is not prime is that it
+        divides the product
+        (2*y-(1+a))*(2*y-(1-a)) = 4*y^2 - 4*y - 4.
+
+        Unfortunately, the program above is already unable to determine
+        that the modulus is irreducible.
+        """
+        from sage.categories.all import IntegralDomains
+        if self.category().is_subcategory(IntegralDomains()):
+            return True
+        ret = self.base_ring().is_integral_domain(proof)
+        if ret:
+            try:
+                irr = self.modulus().is_irreducible()
+                if not irr:
+                    # since the modulus is nonzero, the condition of the base ring being an
+                    # integral domain and the modulus being irreducible are
+                    # necessary but not sufficient
+                    ret = False
+                else:
+                    from sage.categories.gcd_domains import GcdDomains
+                    if self.base_ring() in GcdDomains():
+                        # if the base ring is a GCD domain, the conditions are sufficient
+                        ret = True
+                    else:
+                        raise NotImplementedError
+            except NotImplementedError:
+                if proof:
+                    raise
+                else:
+                    ret = False
+
+        if ret:
+            self._refine_category_(IntegralDomains())
+        return ret
+
     def krull_dimension(self):
         """
         Return the Krull dimension.

src/sage/rings/ring.pyx

@@ -982,11 +982,9 @@ cdef class Ring(ParentWithGens):
             sage: R.fraction_field()
             Traceback (most recent call last):
             ...
-            NotImplementedError
+            TypeError: self must be an integral domain.
             sage: R.is_integral_domain()
-            Traceback (most recent call last):
-            ...
-            NotImplementedError
+            False
 
         Forward the proof flag to ``is_field``, see :trac:`22910`::