sage.rings: Update # needs

Author: Matthias Koeppe

src/sage/rings/number_field/number_field_ideal.py

@@ -1002,6 +1002,7 @@ def is_maximal(self):
 
         EXAMPLES::
 
+            sage: x = polygen(ZZ)
             sage: K.<a> = NumberField(x^3 + 3); K
             Number Field in a with defining polynomial x^3 + 3
             sage: K.ideal(5).is_maximal()
@@ -1017,6 +1018,7 @@ def is_prime(self):
 
         EXAMPLES::
 
+            sage: x = polygen(ZZ)
             sage: K.<a> = NumberField(x^2 - 17); K
             Number Field in a with defining polynomial x^2 - 17
             sage: K.ideal(5).is_prime()   # inert prime
@@ -1031,7 +1033,7 @@ def is_prime(self):
         Check that we do not factor the norm of the ideal, this used
         to take half an hour, see :trac:`33360`::
 
-            sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
+            sage: K.<a,b,c> = NumberField([x^2 - 2, x^2 - 3, x^2 - 5])
             sage: t = (((-2611940*c + 1925290/7653)*b - 1537130/7653*c
             ....:       + 10130950)*a + (1343014/7653*c - 8349770)*b
             ....:       + 6477058*c - 2801449990/4002519)
@@ -1112,6 +1114,7 @@ def _cache_bnfisprincipal(self, proof=None, gens=False):
 
         Check that no warnings are triggered from PARI/GP (see :trac:`30801`)::
 
+            sage: x = polygen(ZZ)
             sage: K.<a> = NumberField(x^2 - x + 112941801)
             sage: I = K.ideal((112941823, a + 49942513))
             sage: I.is_principal()
@@ -1494,7 +1497,7 @@ def decomposition_group(self):
 
         EXAMPLES::
 
-            sage: QuadraticField(-23, 'w').primes_above(7)[0].decomposition_group()
+            sage: QuadraticField(-23, 'w').primes_above(7)[0].decomposition_group()     # needs sage.groups
             Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)
         """
         return self.number_field().galois_group().decomposition_group(self)
@@ -1510,9 +1513,9 @@ def ramification_group(self, v):
 
         EXAMPLES::
 
-            sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(0)
+            sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(0)    # needs sage.groups
             Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)
-            sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(1)
+            sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(1)    # needs sage.groups
             Subgroup generated by [()] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)
         """
 
@@ -1528,7 +1531,7 @@ def inertia_group(self):
 
         EXAMPLES::
 
-            sage: QuadraticField(-23, 'w').primes_above(23)[0].inertia_group()
+            sage: QuadraticField(-23, 'w').primes_above(23)[0].inertia_group()          # needs sage.groups
             Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)
         """
         return self.ramification_group(0)
@@ -1594,7 +1597,7 @@ def artin_symbol(self):
 
         EXAMPLES::
 
-            sage: QuadraticField(-23, 'w').primes_above(7)[0].artin_symbol()
+            sage: QuadraticField(-23, 'w').primes_above(7)[0].artin_symbol()            # needs sage.groups
             (1,2)
         """
         return self.number_field().galois_group().artin_symbol(self)

src/sage/rings/number_field/totallyreal_rel.py

@@ -51,7 +51,7 @@
     sage: [ f[0] for f in ls ]
     [725, 1125, 1600, 2000, 2225, 2525, 3600, 4225, 4400, 4525, 5125, 5225, 5725, 6125, 7225, 7600, 7625, 8000, 8525, 8725, 9225]
 
-    sage: [NumberField(ZZx(x[1]), 't').is_galois() for x in ls]
+    sage: [NumberField(ZZx(x[1]), 't').is_galois() for x in ls]                         # needs sage.groups
     [False, True, True, True, False, False, True, True, False, False, False, False, False, True, True, False, False, True, False, False, False]
 
 Eight out of 21 such fields are Galois (with Galois group `C_4`