Ticket #10772: trac_10772-0pow0.patch

sage/matrix/matrix0.pyx

@@ -4105 +4105 @@
             [      -3/11     -13/121   1436/1331]
             [    127/121   -337/1331 -4445/14641]
             [    -13/121   1436/1331 -8015/14641]
+
+        Sage follows Python's convention 0^0 = 1, as each of the following
+        examples show::
+
+            sage: a = Matrix([[1,0],[0,0]]); a
+            [1 0]
+            [0 0]
+            sage: a^0 # lower right entry is 0^0
+            [1 0]
+            [0 1]
+            sage: Matrix([[0]])^0
+            [1]
+            sage: 0^0
+            1
         """
         if not self.is_square():
             raise ArithmeticError("self must be a square matrix")

sage/rings/number_field/number_field.py

@@ -6129 +6129 @@
             sage: L.base_field().base_field()
             Rational Field
 
+            sage: L = K.relativize(K(0), 'a'); L
+            Number Field in a0 with defining polynomial x^4 + 2*x^2 + 2 over its base field
+            sage: L.base_field()
+            Number Field in a1 with defining polynomial x
+            sage: L.base_field().base_field()
+            Rational Field
+
+        We can relativize over morphisms returned by self.subfields()::
+
+            sage: L = NumberField(x^4 + 1, 'a')
+            sage: [L.relativize(h, 'c') for (f,h,i) in L.subfields()]
+            [Number Field in c0 with defining polynomial x^4 + 1 over its base field, Number Field in c0 with defining polynomial x^2 - 1/2*a1 over its base field, Number Field in c0 with defining polynomial x^2 - a2*x - 1 over its base field, Number Field in c0 with defining polynomial x^2 - a3*x + 1 over its base field, Number Field in c0 with defining polynomial x - a4 over its base field]
+
         We can relativize over a relative field::
 
             sage: K.<z> = CyclotomicField(16)

sage/rings/number_field/number_field_element.pyx

@@ -1257 +1257 @@
             sage: sqrt2^sqrt2
             2^(1/2*sqrt(2))
 
+        Sage follows Python's convention 0^0 = 1::
+
+            sage: a = K(0)^0; a
+            1
+            sage: a.parent()
+            Number Field in sqrt2 with defining polynomial x^2 - 2
+
         TESTS::
 
             sage: 2^I

sage/rings/polynomial/polydict.pyx

@@ -722 +722 @@
             sage: f**0
             PolyDict with representation {(0, 0): 1}
             sage: (f-f)**0
-            Traceback (most recent call last):
-            ...
-            ArithmeticError: 0^0 is undefined.
+            PolyDict with representation {0: 1}
         """
         return generic_power(self, n, self.__one())
 

sage/structure/element.pyx

@@ -2888 +2888 @@
         except TypeError:
             raise NotImplementedError, "non-integral exponents not supported"
 
-    # It's better to test first the equality to zero of the exponent. Indeed,
-    # the following tests tree allows us be sure that "non a" is never called
-    # if n is not 0, so that we can deals with this particular case before
-    # calling generic_power. It is needed to handle the case of semi-groups
-    # (without unit) where "not a" does not makes sense and there is no one to
-    # return when n is 0.
     if not n:
-        if not a:
-            raise ArithmeticError, "0^0 is undefined."
         if one is None:
             if PY_TYPE_CHECK(a, Element):
                 return (<Element>a)._parent(1)