Trac #18942: Weird bug in roots of a polynomial in relative number fi…

…eld extension

I have no idea what is going on.

{{{
sage: F.<omega> = NumberField(x^2+x+1)
sage: xx = polygen(F)
sage: ABs = []
sage: ps = [p for p, _ in F(7).factor()]
sage: for mu in ps:
    K = F.extension(xx^3 - mu, 'alpha')
    print K.defining_polynomial().roots(K)
sage: for mu in ps:
    K = F.extension(xx^3 - mu, 'alpha')
    print K.defining_polynomial().roots(K)

[(alpha, 1), ((-omega - 1)*alpha, 1), (omega*alpha, 1)]
[(alpha, 1), (omega*alpha, 1), ((-omega - 1)*alpha, 1)]
[]
[(alpha, 1), (omega*alpha, 1), ((-omega - 1)*alpha, 1)]
}}}

So, that's weird. But it gets worse! First do this

{{{
sage: fbar = xx^3 - ps[0]
sage: Kbar = F.extension(fbar, 'alpha')
sage: fbar.roots(Kbar)
[]
}}}

Okay, but then do fbar.roots?? to see the source code, then press 'q' to
exit that, then

{{{
sage: fbar.roots(Kbar)
[(alpha, 1), ((-omega - 1)*alpha, 1), (omega*alpha, 1)]
}}}

Huh?

(I'm doing this is sage 6.7 on the cloud.)

URL: http://trac.sagemath.org/18942
Reported by: robharron
Ticket author(s): Kiran Kedlaya
Reviewer(s): Peter Bruin

src/sage/rings/number_field/number_field.py

@@ -25,6 +25,8 @@
 
 - Vincent Delecroix (2015-02): comparisons/floor/ceil using embeddings
 
+- Kiran Kedlaya (2016-05): relative number fields hash based on relative polynomials
+
 .. note::
 
    Unlike in PARI/GP, class group computations *in Sage* do *not* by default
@@ -2719,6 +2721,19 @@ def __cmp__(self, other):
 
             sage: M['x'] == L['x']
             False
+            
+        Similarly, two relative number fields which are isomorphic as absolute
+        fields, but which are not presented the same way, should not be equal (see
+        :trac:`18942`)::
+
+            sage: F.<omega> = NumberField(x^2 + x + 1)
+            sage: y = polygen(F)
+            sage: K = F.extension(y^3 + 3*omega + 2, 'alpha')
+            sage: L = F.extension(y^3 - 3*omega - 1, 'alpha')
+            sage: K == L
+            False
+            sage: K.is_isomorphic(L)
+            True
 
         """
         if self is other:
@@ -2730,7 +2745,7 @@ def __cmp__(self, other):
         c = cmp(self.variable_name(), other.variable_name())
         if c: return c
         # compare coefficients so that the polynomial variable does not count
-        c = cmp(list(self.__polynomial), list(other.__polynomial))
+        c = cmp(list(self.relative_polynomial()), list(other.relative_polynomial()))
         if c: return c
         # Now we compare the embeddings (if any).
         f, g = self.coerce_embedding(), other.coerce_embedding()
@@ -2763,8 +2778,47 @@ def __hash__(self):
             sage: hash(K) == hash(L)
             True
 
-        """
-        return hash((self.variable_name(), self.base_field(), tuple(self.__polynomial)))
+        The number fields ``K`` and ``L`` in the following example used to have
+        the same hash prior to :trac:`18942`::
+        
+            sage: F.<omega> = NumberField(x^2 + x + 1)
+            sage: y = polygen(F)
+            sage: K = F.extension(y^3 + 3*omega + 2, 'alpha')
+            sage: L = F.extension(y^3 - 3*omega - 1, 'alpha')
+            sage: K == L
+            False
+            sage: K.is_isomorphic(L)
+            True
+            sage: hash(K) == hash(L)
+            False
+            
+        This example illustrates the issue resolved in :trac:`18942`::
+        
+            sage: F.<omega> = NumberField(x^2+x+1)
+            sage: xx = polygen(F)
+            sage: ps = [p for p, _ in F(7).factor()]
+            sage: for mu in ps:
+            ....:     K = F.extension(xx^3 - mu, 'alpha')
+            ....:     print(K.defining_polynomial().roots(K))
+            [(alpha, 1), ((-omega - 1)*alpha, 1), (omega*alpha, 1)]
+            [(alpha, 1), (omega*alpha, 1), ((-omega - 1)*alpha, 1)]
+            sage: for mu in ps:
+            ....:     K = F.extension(xx^3 - mu, 'alpha')
+            ....:     print(K.defining_polynomial().roots(K))
+            [(alpha, 1), ((-omega - 1)*alpha, 1), (omega*alpha, 1)]
+            [(alpha, 1), (omega*alpha, 1), ((-omega - 1)*alpha, 1)]
+            
+        This example was suggested on sage-nt; see :trac:`18942`::
+
+            sage: G=DirichletGroup(80);
+            sage: for chi in G:
+            ....:     D=ModularSymbols(chi,2,-1).cuspidal_subspace().new_subspace().decomposition()
+            ....:     for f in D:
+            ....:         elt=f.q_eigenform(10,'alpha')[3];
+            ....:         assert(elt.is_integral())
+
+        """
+        return hash((self.variable_name(), self.base_field(), tuple(self.relative_polynomial())))
 
     def _ideal_class_(self, n=0):
         """