Cannot create homomorphism from prime residue field to finite field

[johanbosman]

Consider K = QQ(sqrt(337)). The prime 5 is inert in K and the prime 13 splits. We'll start with 13 to see what goes wrong:

sage: K.<w> = QuadraticField(337)
sage: pp = K.ideal(13).factor()[0][0]
sage: RF13 = K.residue_field(pp)
sage: RF13.gens()
(1,)
sage: RF13.hom([GF(13)(1)])
...
TypeError: images do not define a valid homomorphism

However, for residue fields that aren't prime fields it does work:

sage: RF5 = K.residue_field(5)
sage: F25 = GF(25, names='a', modulus=RF5.polynomial())
sage: RF5.hom([F25.0])
Ring morphism:
  From: Residue field in wbar of Fractional ideal (5)
  To:   Finite Field in a of size 5^2
  Defn: wbar |--> a

Btw, this is in sage 4.6.2. on Mac OS 10.6.7.

[aly.deines]

I was trying out the patch, and the following example didn't work out like I expected:

sage: K
Number Field in a with defining polynomial x^3 - 2
sage: K.<a> = NumberField(x^3-2)
sage: P = K.ideal(7)
sage: P.factor()
Fractional ideal (7)
sage: k = P.residue_field()
sage: k.order()
343
sage: F.<b> = GF(343)
sage: F.order()
343
sage: k
Residue field in abar of Fractional ideal (7)
sage: F
Finite Field in b of size 7^3
sage: k.hom([F(1)])
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)

/Users/aly/Desktop/sage-4.7.rc2/<ipython console> in <module>()

/Users/aly/Desktop/sage-4.7.rc2/local/lib/python2.6/site-packages/sage/structure/parent_gens.so in sage.structure.parent_gens.ParentWithGens.hom (sage/structure/parent_gens.c:3792)()

/Users/aly/Desktop/sage-4.7.rc2/local/lib/python2.6/site-packages/sage/structure/parent.so in sage.structure.parent.Parent.hom (sage/structure/parent.c:9067)()

/Users/aly/Desktop/sage-4.7.rc2/local/lib/python2.6/site-packages/sage/rings/finite_rings/homset.pyc in __call__(self, im_gens, check)
     49                 return self._coerce_impl(im_gens)
     50             except TypeError:
---> 51                 raise TypeError, "images do not define a valid homomorphism"
     52         
     53     def _coerce_impl(self, x):

TypeError: images do not define a valid homomorphism   
sage: k.has_coerce_map_from(F)
False

Should this patch fix this as well?

[johanbosman]

In your example, you're trying to send the generator of k to the element 1 of F, which does indeed not define a field homomorphism. This does work:

sage: root = k.polynomial().roots(F)[0][0]
sage: k.hom([root])
Ring morphism:
  From: Residue field in abar of Fractional ideal (7)
  To:   Finite Field in b of size 7^3
  Defn: abar |--> 4*b^2 + 4*b + 2

[kedlaya]

Looks fine, applies against 4.7, no long doctest failures.

[jdemeyer]

I think the new patch is a bit low on examples. For example, the ticket mentions number fields, so why not add a number field example?

[johanbosman]

To be used on top of the first patch

[johanbosman]

Example added. ;)