Creating homomorphisms of relative number fields seems totally broken

[williamstein]

The following _should_ create the correct homomorphism (complex conjugation, see #4725):

sage: K.<j,b> = QQ[sqrt(-1), sqrt(2)]
sage: conj = K.hom([-j, b])
boom!

However it doesn't.

CC: @lftabera

Component: number fields

Author: Robert Harron

Reviewer: Francis Clarke

Issue created by migration from https://trac.sagemath.org/ticket/4726

[williamstein]

Description changed:

--- 
+++ 
@@ -1,4 +1,4 @@
-The following `_should_` create the correct homomorphism (complex conjugation, see #4724):
+The following `_should_` create the correct homomorphism (complex conjugation, see #4725):
 
 ```
 sage: K.<j,b> = QQ[sqrt(-1), sqrt(2)]

[rharron]

Author: Robert Harron

[rharron]

comment:3

I just uploaded a patch that allows you to construct relative number field morphisms as above (the code simply wasn't written accept something like that). I just made it go recursively through the list you provide, so in a tower of fields, you only need to provide the images up to the point where your morphism is the identity.

I'll set this to needs review after I do a full doctest overnight (the number_field module passes all doctests).

[sagetrac-fwclarke]

comment:5

This mostly works well. I tried it on some cases where the relative
degrees were different. For example

sage: C.<z> = CyclotomicField(15)
sage: K = C.relativize(z^5 + 1, 'a'); K
Number Field in a0 with defining polynomial x^4 - a1*x^3 + (a1 - 1)*x^2 + x - a1 over its base field
sage: K.inject_variables()
Defining a0, a1
sage: L = C.relativize(z^4 + z, 'b'); L
Number Field in b0 with defining polynomial x^2 - b1*x - 1/2*b1^3 + b1^2 - b1 - 1/2 over its base field
sage: H = Hom(K, L)
sage: K.hom(map(H[2], K.gens()))
Relative number field morphism:
  From: Number Field in a0 with defining polynomial x^4 - a1*x^3 + (a1 - 1)*x^2 + x - a1 over its base field
  To:   Number Field in b0 with defining polynomial x^2 - b1*x - 1/2*b1^3 + b1^2 - b1 - 1/2 over its base field
  Defn: a0 |--> -b0 + b1
        a1 |--> -1/2*b1^3 + b1^2 - b1 + 1/2

My concerns are about when the default base homomorphism is used. It is certainly not correct to describe default_base_hom as "trivial", and not clear anyway what that would mean if the domain and codomain differ. Its docstring says

Pick an embedding of the base field of self into the codomain of this homset. This is done in an essentially arbitrary way.

Since the value is cached, see line 526 of sage/rings/number_field/morphism.py (after your patch is applied), using the default argument base_hom=None should always give the same restriction to the base_field. However, continuing the above example,

sage: [K.hom([h(a0)])(a1) for h in H]
[-1/2*b1^3 + b1^2 - b1 + 1/2,
 1/2*b1^3 - b1^2 + b1 + 1/2,
 -1/2*b1^3 + b1^2 - b1 + 1/2,
 1/2*b1^3 - b1^2 + b1 + 1/2,
 -1/2*b1^3 + b1^2 - b1 + 1/2,
 1/2*b1^3 - b1^2 + b1 + 1/2,
 1/2*b1^3 - b1^2 + b1 + 1/2,
 -1/2*b1^3 + b1^2 - b1 + 1/2]

This happens because
sage.rings.number_field.morphism.RelativeNumberFieldHomset._from_im doesn't do what it claims. With again the same definitions:

sage: b0 = L.gen(); b1 = L.base_field().gen()
sage: base_hom = K.base_field().hom([1/2*b1^3 - b1^2 + b1 + 1/2]); base_hom
Ring morphism:
  From: Number Field in a1 with defining polynomial x^2 - x + 1
  To:   Number Field in b1 with defining polynomial x^4 - x^3 + 2*x^2 + x + 1
  Defn: a1 |--> 1/2*b1^3 - b1^2 + b1 + 1/2
sage: H._from_im([b0], base_hom)
Relative number field morphism:
  From: Number Field in a0 with defining polynomial x^4 - a1*x^3 + (a1 - 1)*x^2 + x - a1 over its base field
  To:   Number Field in b0 with defining polynomial x^2 - b1*x - 1/2*b1^3 + b1^2 - b1 - 1/2 over its base field
  Defn: a0 |--> b0
        a1 |--> -1/2*b1^3 + b1^2 - b1 + 1/2

which does not restrict to the base_field correctly. I note that at present _from_im is not used anywhere else in Sage.

In fact since

sage: K.absolute_generator()
a0

there is only one homomorphism from K to L that sends a0 to b0, so an error should have been raised by H._from_im([b0], base_hom).