#10677: Improve PARI interface for relative number fields
------------------------------+---------------------------------------------
   Reporter:  jdemeyer        |       Owner:  davidloeffler
       Type:  enhancement     |      Status:  needs_review 
   Priority:  major           |   Milestone:  sage-4.6.2   
  Component:  number fields   |    Keywords:  pari rnf     
     Author:  Jeroen Demeyer  |    Upstream:  N/A          
   Reviewer:                  |      Merged:               
Work_issues:                  |  
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Comment(by mstreng):

 Replying to [comment:7 jdemeyer]:
 > The following is a proper example of a PARI relative number field
 element::
 > {{{
 > Mod(Mod(y, y^3 + 17)*x + Mod(1, y^3 + 17), x^2 - y)
 > }}}

 Of course, sorry about that.

 I don't know how much work it is, but as you are adding extra examples
 anyway, perhaps you could fix this:
 {{{
 sage: K.<a> = QuadraticField(-5)
 sage: p = pari("Mod(x,x^2+7)")
 sage: K(p)
 a
 sage: K(p)^2
 -5
 sage: p^2
 Mod(-7, x^2 + 7)
 }}}
 The same issue is there with your code:
 {{{
 sage: y = PolynomialRing(QQ,'y').gen()
 sage: K.<a> = NumberField(y^3+17)
 sage: x = PolynomialRing(K,'x').gen()
 sage: L.<b> = NumberField(x^2-a)
 sage: q = pari("Mod(Mod(y, y^3 + 19)*x + Mod(1, y^3 + 23), x^2 - y + 1)")
 sage: L(q)
 a*b
 }}}
 I think K(p) and L(q) should raise errors.

 > There is an example in my patch in
 `sage/rings/number_field/number_field_element.pyx`, line 3316.  But it
 might be a good idea to expand this.

 Great! I'd say

 * at least one example with the completely written out correct form of
 PARI, in case idiots like me swap the moduli around again.

 * at least one example with the incorrect moduli that raises an error
 (like my examples in this comment).

 * at least one example in the documentation of the {{{__init__}}} method
 of {{{NumberFieldElement}}} (because that is where your new code is).

-- 
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10677#comment:10>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica, 
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