Fix multiplication of number field element * ZZ-vector by handling latex names of number field generators

[kliem]

sage: 1/2*vector([1,2,3])
(1/2, 1, 3/2)

sage: AA(2).sqrt()*vector([1,2,3])                                                                                                                                                                                                                                                                                                                                         
(1.414213562373095?, 2.828427124746190?, 4.242640687119285?)

sage: K.<sqrt5> = QuadraticField(5, embedding=AA(5).sqrt())                                                                                                                                                                                                                                                                                                                
sage: sqrt5*vector([1,2])                                                                                                                                                                                                                                                                                                                                                  
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-65-1ac790b433d6> in <module>
----> 1 sqrt5*vector([Integer(1),Integer(2)])

~/Applications/sage/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12196)()
   1513             return (<Element>left)._mul_(right)
   1514         if BOTH_ARE_ELEMENT(cl):
-> 1515             return coercion_model.bin_op(left, right, mul)
   1516 
   1517         cdef long value

~/Applications/sage/local/lib/python3.7/site-packages/sage/structure/coerce.pyx in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:11304)()
   1246         # We should really include the underlying error.
   1247         # This causes so much headache.
-> 1248         raise bin_op_exception(op, x, y)
   1249 
   1250     cpdef canonical_coercion(self, x, y):

TypeError: unsupported operand parent(s) for *: 'Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?' and 'Ambient free module of rank 2 over the principal ideal domain Integer Ring'

It doesn't matter whether you specify the embedding or not.

Depends on #30372

CC: @videlec @tscrim

Component: coercion

Keywords: action, scalar, free module

Author: Matthias Koeppe, Francis Clarke

Branch/Commit: a3c790f

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/30360

[mkoeppe]

comment:2

It's not really the pushout that's failing here:

sage: from sage.categories.pushout import pushout
sage: pushout(sqrt5.parent(), vector([1,2]).parent())
Vector space of dimension 2 over Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?

[mkoeppe]

comment:3

I think the methods _lmul_ and _rmul_ of NumberFieldElement_quadratic are in the way. I think they may just need to be removed (haven't tried)

[kliem]

Changed keywords from pushout, free module to action, scalar, free module

[mkoeppe]

Branch: u/mkoeppe/action_of_number_field_on_free_module_over_zz_not_discovered

[mkoeppe]

Commit: 8a9c2aa

[mkoeppe]

comment:7

Replying to @mkoeppe:

I think the methods _lmul_ and _rmul_ of NumberFieldElement_quadratic are in the way. I think they may just need to be removed (haven't tried)

By itself this does not fix it... (see branch)

---

New commits:

8a9c2aa

WIP: NumberFieldElement_quadratic._lmul_, _rmul_: Remove

[mkoeppe]

comment:8

You are right, the action is missing.

sage: cm.explain(AA, vector([1,2]).parent(), operator.mul)                                                                                                          
Action discovered.
    Left scalar multiplication by Algebraic Real Field on Ambient free module of rank 2 over the principal ideal domain Integer Ring
Result lives in Vector space of dimension 2 over Algebraic Real Field
Vector space of dimension 2 over Algebraic Real Field
sage: cm.explain(K, vector([1,2]).parent(), operator.mul)                                                                                                           

Unknown result parent.

[mkoeppe]

comment:9

It looks like this is because of a unique representation / equality bug for number fields

[mkoeppe]

comment:10

sage: K.<sqrt5> = QuadraticField(5, embedding=AA(5).sqrt())
sage: V = vector([1, 2]).parent()
sage: from sage.structure.coerce_actions import LeftModuleAction                                                                                                    
sage: LeftModuleAction(AA, V)
Left scalar multiplication by Algebraic Real Field on Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: LeftModuleAction(K, V)                                                                                                                               
CoercionException: Actor must be coercible into base.

Looking at LeftModuleAction.__init__, this boils down to:

sage: from sage.categories.pushout import pushout                                                         
sage: K                                                                                                   
Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?
sage: pushout(K, ZZ) is K                                                                                 
True
sage: K2 = pushout(K, V).base(); K2                                                                       
Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?
sage: K2 is K                                                                                             
False
sage: K2 == K                                                                                             
False