#19171: New method divides

Author: bgrenet

src/sage/rings/polynomial/polynomial_element.pyx

@@ -8496,6 +8496,48 @@ cdef class Polynomial(CommutativeAlgebraElement):
             raise ValueError("not a %s power"%m.ordinal_str())
 
 
+    def divides(self, p):
+        r"""
+        Return `True` if this polynomial divides `p`.
+
+        EXAMPLES::
+
+            sage: R.<x> = ZZ[]
+            sage: (2*x + 1).divides(4*x**2 - 1)
+            True
+            sage: (2*x + 1).divides(4*x**2 + 1)
+            False
+            sage: (2*x + 1).divides(R(0))
+            True
+            sage: R(0).divides(2*x + 1)
+            False
+            sage: R(0).divides(R(0))
+            True
+            sage: R.<x> = PolynomialRing(ZZ, implementation="NTL")
+            sage: (2*x + 1).divides(4*x**2 + 1)
+            False
+        """
+        parent = self.parent()
+        if p.parent() is parent:
+            if p.is_zero(): return True          # everything divides 0
+            if self.is_zero(): return False      # 0 only divides 0
+            try:
+                if self.is_unit(): return True   # units divide everything
+            except NotImplementedError:
+                pass
+            if self.is_one(): return True        # if is_unit is not implemented
+
+            try:
+                return (p % self) == 0           # if quo_rem is defined
+            except ArithmeticError:
+                return False                     # if division is not exact
+
+        else:
+            cm = sage.structure.element.get_coercion_model()
+            a, b = cm.canonical_coercion(self, p)
+            return a.divides(b)
+
+
 # ----------------- inner functions -------------
 # Cython can't handle function definitions inside other function