diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index 63a80c0d0ca..287464d593c 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -2616,10 +2616,8 @@ def uniformizer(self, P, others = "positive"):
             [True, True, True]
             sage: [pi.valuation(P) for pi,P in zip(pilist,Plist)]
             [1, 1, 1]
-            sage: [Plist[i]==K.ideal(pilist[i]) for i in range(len(Plist))]
-            [True, False, True]
-
-
+            sage: [ pilist[i] in Plist[i] for i in range(len(Plist)) ]
+            [True, True, True]
         """
         if not is_NumberFieldIdeal(P):
             P = self.ideal(P)
diff --git a/src/sage/rings/number_field/number_field_ideal.py b/src/sage/rings/number_field/number_field_ideal.py
index 7e251ddd81c..52223de5617 100644
--- a/src/sage/rings/number_field/number_field_ideal.py
+++ b/src/sage/rings/number_field/number_field_ideal.py
@@ -864,9 +864,11 @@ def prime_factors(self):
 
         EXAMPLES:
             sage: K.<w> = NumberField(x^2 + 23)
-            sage: I = ideal(w*23^5)
+            sage: I = ideal(w+1)
             sage: I.prime_factors()
-            [Fractional ideal (-w)]
+            [Fractional ideal (2, 1/2*w - 1/2),
+            Fractional ideal (2, 1/2*w + 1/2),
+            Fractional ideal (3, -1/2*w - 1/2)]
         """
         return [x[0] for x in self.factor()]