Avoid nfinit while factoring polynomials

[lftabera]

In previous versions of pari the options to factor a univariate polynomial over a number field were Trager's or Van Hoeij modular algorithm. The second method is the preferred one, but it used to need a nf structure.

Hence Sage computed nfinit on the number field before factoring the polynomial via Pari.

With current pari version the whole nf structure is not needed. So, the factor routines should not call nfinit that can be a very expensive operation for large fields.

The patch modifies the factor method. If the nf structure is already computed we use it, as it will be faster. If the nf structure is not already computed then do not compute it to factor the polynomial.

See also #11890 for a slightly different solution to the same issue.

[lftabera]

The doctest failure is known bug that is corrected in #2329.

Depends: #2329

[mariah]

Minor edit of grammer.

[mariah]

The patch applied to sage-4.7.rc2 causes the following test to fail:

% ./sage -t  -long -force_lib "devel/sage/sage/rings/polynomial/polynomial_element.pyx"
sage -t -long -force_lib "devel/sage/sage/rings/polynomial/polynomial_element.pyx"
**********************************************************************
File "/home/mariah/sage/sage-4.7.rc2-x86_64-Linux-nehalem-fc-review-10910/devel/sage/sage/rings/polynomial/polynomial_element.pyx", line 2704:
    sage: hasattr(N, '_NumberField_generic__pari_nf')
Expected:
    False
Got:
    True
**********************************************************************
1 items had failures:
   1 of 143 in __main__.example_51
***Test Failed*** 1 failures.
For whitespace errors, see the file /home/mariah/.sage//tmp/.doctest_polynomial_element.py
         [23.1 s]
 
----------------------------------------------------------------------
The following tests failed:


        sage -t -long -force_lib "devel/sage/sage/rings/polynomial/polynomial_element.pyx"
Total time for all tests: 23.1 seconds

[lftabera]

Just for the record, my patch introduces the following problem, so I need to add a doctest to it since it is an unapparent side effect.

sage: N = NumberField(x^3-2*x+3,'a')
sage: f = N[x](x^2-41943301)
sage: M = N.extension(f,'b')
sage: M.hom([M.gen()])
...
ValueError: no way to map base field to codomain.

[lftabera]

I have tracked the latest problem and it is in fact a bug in pari. See upstream report ​#1207.

I have corrected some errors in the code and added another doctest.

I am unable to reproduce the doctest failing that Maria points out on sage-4.7

Note: doctest with this patch will fail until the pari bug is solved.

[lftabera]

The problem with pari is solved in the last stable release.

This ticket now depends on #11130 until pari is not updated the patch is not safe.

[lftabera]

the good one

[lftabera]

Rebase against 4.7.1

[lftabera]

This ticked should be closed as a duplicate of #11890