trac #15390: roots and eigenvalues over finite fields
trac #15390: merge sage 6.3.beta4
trac #15390: fix the case of polynomial over algebraic closure
trac #15390: make factorization works as well
trac 15390: put space after comma
Added doctest for #12410.
Trac #15390: roots of polynomials and eigenvalues of matrices over finite fields

Based on #14990 we implement root computations in the algebraic closure
of finite fields.

It should be nice to also adapt eigenvectors... (but we need before
algebraic conjugates)

Currently finite subfields of an algebraic closure can not talk to each
other (see in particular #16509). But hopefully, it is possible to
compute roots without it!

URL: http://trac.sagemath.org/15390
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix
Reviewer(s): Miguel Marco
Trac #12410: divergent alternating sum gives finite result

The two sums
{{{
sage: var('x');
sage: sum(1/x*(-1)^x, x, 1, oo)
-log(2)
sage: sum(1/x*(-1)^x, x, 0, oo)
-log(2)
}}}
return the same, but shouldn't. The latter sum is divergent because of
the summand `1/0`.

It works correctly, when not using the `(-1)^x`, e.g.:
{{{
sage: sum(1/x^2, x, 1, oo)
1/6*pi^2
sage: sum(1/x^2, x, 0, oo)
Traceback (most recent call last)
...
ValueError: Sum is divergent.
}}}
(Calculating `sum(1/x^2*(-1)^x, x, 0, oo)` gives the wrong result, too.)

URL: http://trac.sagemath.org/12410
Reported by: dkrenn
Ticket author(s): Travis Scrimshaw
Reviewer(s): Peter Bruin