trac #17671: fix xgcd of polynomial_field_generic

As a consequence we can revert some of the modifications from
function_field.py.

src/sage/categories/principal_ideal_domains.py

@@ -70,6 +70,7 @@ def _test_gcd_vs_xgcd(self, **options):
                 sage: ZZ._test_gcd_vs_xgcd()
                 sage: QQ._test_gcd_vs_xgcd()
                 sage: QQ['x']._test_gcd_vs_xgcd()
+                sage: QQbar['x']._test_gcd_vs_xgcd()
 
             A slightly more involved example of polynomial ring with a non UFD
             base ring::

src/sage/rings/function_field/function_field.py

@@ -66,8 +66,8 @@
 It is not the fault of the function field code so this will
 be fixed in another ticket::
 
-    sage: TestSuite(R).run(skip = ('_test_elements', '_test_gcd_vs_xgcd'))
-    sage: TestSuite(S).run(skip = ('_test_elements', '_test_gcd_vs_xgcd'))
+    sage: TestSuite(R).run(skip = '_test_elements')
+    sage: TestSuite(S).run(skip = '_test_elements')
 """
 #*****************************************************************************
 #       Copyright (C) 2010 William Stein <wstein@gmail.com>

src/sage/rings/polynomial/polynomial_element_generic.py

@@ -732,6 +732,13 @@ def gcd(self, other):
             sage: (2*x).gcd(2*x^2)
             x
 
+            sage: zero = R.zero_element()
+            sage: zero.gcd(2*x)
+            x
+            sage: (2*x).gcd(zero)
+            x
+            sage: zero.gcd(zero)
+            0
         """
         from sage.categories.euclidean_domains import EuclideanDomains
         g = EuclideanDomains().ElementMethods().gcd(self, other)
@@ -774,25 +781,43 @@ def xgcd(self, other):
             sage: g == u*P(0) + v*x
             True
 
+        TESTS:
+
+        We check that the behavior of xgcd with zero elements is compatible with
+        gcd (:trac:`17671`)::
+
+            sage: R.<x> = QQbar[]
+            sage: zero = R.zero_element()
+            sage: zero.xgcd(2*x)
+            (x, 0, 1/2)
+            sage: (2*x).xgcd(zero)
+            (x, 1/2, 0)
+            sage: zero.xgcd(zero)
+            (0, 0, 0)
         """
+        R = self.parent()
+        zero = R.zero_element()
+        one = R.one_element()
         if other.is_zero():
-            R = self.parent()
-            return self, R.one_element(), R.zero_element()
+            if self.is_zero():
+                return (zero, zero, zero)
+            else:
+                c = self.leading_coefficient()
+                return (self/c, one/c, zero)
+        elif self.is_zero():
+            c = other.leading_coefficient()
+            return (other/c, zero, one/c)
+
         # Algorithm 3.2.2 of Cohen, GTM 138
-        R = self.parent()
         A = self
         B = other
-        U = R.one_element()
+        U = one
         G = A
-        V1 = R.zero_element()
+        V1 = zero
         V3 = B
         while not V3.is_zero():
             Q, R = G.quo_rem(V3)
-            T = U - V1*Q
-            U = V1
-            G = V3
-            V1 = T
-            V3 = R
+            G, U, V1, V3 = V3, V1, U-V1*Q, R
         V = (G-A*U)//B
         lc = G.leading_coefficient()
         return G/lc, U/lc, V/lc