Trac 23851: accept both roots of unity of order 6 in doctests

src/sage/rings/polynomial/polynomial_quotient_ring.py

@@ -1386,37 +1386,36 @@ def S_units(self, S, proof=True):
             sage: K.unit_group()
             Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
             sage: K.<a> = QQ['x'].quotient(x^2 + 3)
-            sage: u,o = K.S_units([])[0]; u, o
-            (-1/2*a + 1/2, 6)
+            sage: u,o = K.S_units([])[0]; o
+            6
+            sage: 2*u - 1 in {a, -a}
+            True
             sage: u^6
             1
             sage: u^3
             -1
-            sage: u^2
-            -1/2*a - 1/2
+            sage: 2*u^2 + 1 in {a, -a}
+            True
 
         ::
 
             sage: K.<a> = QuadraticField(-3)
             sage: y = polygen(K)
             sage: L.<b> = K['y'].quotient(y^3 + 5); L
             Univariate Quotient Polynomial Ring in b over Number Field in a with defining polynomial x^2 + 3 with modulus y^3 + 5
-            sage: L.S_units([])
-            [(-1/2*a + 1/2, 6),
-             ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
-             (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
-            sage: L.S_units([K.ideal(1/2*a - 3/2)])
-            [((-1/6*a - 1/2)*b^2 + (1/3*a - 1)*b + 4/3*a, +Infinity),
-             (-1/2*a + 1/2, 6),
-             ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
-             (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
-            sage: L.S_units([K.ideal(2)])
-            [((1/2*a - 1/2)*b^2 + (a + 1)*b + 3, +Infinity),
-             ((1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a + 1/2, +Infinity),
-             ((1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a - 1/2, +Infinity),
-             (-1/2*a + 1/2, 6),
-             ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
-             (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
+            sage: [u for u, o in L.S_units([]) if o is Infinity]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
+             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            sage: [u for u, o in L.S_units([K.ideal(1/2*a - 3/2)]) if o is Infinity]
+            [(-1/6*a - 1/2)*b^2 + (1/3*a - 1)*b + 4/3*a,
+             (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
+             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            sage: [u for u, o in L.S_units([K.ideal(2)]) if o is Infinity]
+            [(1/2*a - 1/2)*b^2 + (a + 1)*b + 3,
+             (1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a + 1/2,
+             (1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a - 1/2,
+             (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
+             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
 
         Note that all the returned values live where we expect them to::
 
@@ -1473,14 +1472,15 @@ def units(self, proof=True):
             sage: K.unit_group()
             Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
             sage: K.<a> = QQ['x'].quotient(x^2 + 3)
-            sage: u = K.units()[0][0]; u
-            -1/2*a + 1/2
+            sage: u = K.units()[0][0]
+            sage: 2*u - 1 in {a, -a}
+            True
             sage: u^6
             1
             sage: u^3
             -1
-            sage: u^2
-            -1/2*a - 1/2
+            sage: 2*u^2 + 1 in {a, -a}
+            True
             sage: K.<a> = QQ['x'].quotient(x^2 + 5)
             sage: K.units(())
             [(-1, 2)]
@@ -1491,17 +1491,16 @@ def units(self, proof=True):
             sage: y = polygen(K)
             sage: L.<b> = K['y'].quotient(y^3 + 5); L
             Univariate Quotient Polynomial Ring in b over Number Field in a with defining polynomial x^2 + 3 with modulus y^3 + 5
-            sage: L.units()
-            [(-1/2*a + 1/2, 6),
-             ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
-             (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
+            sage: [u for u, o in L.units() if o is Infinity]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
+             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
             sage: L.<b> = K.extension(y^3 + 5)
             sage: L.unit_group()
             Unit group with structure C6 x Z x Z of Number Field in b with defining polynomial x^3 + 5 over its base field
             sage: L.unit_group().gens()    # abstract generators
             (u0, u1, u2)
-            sage: L.unit_group().gens_values()
-            [1/2*a + 1/2, (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            sage: L.unit_group().gens_values()[1:]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
 
         Note that all the returned values live where we expect them to::