Implement a generic radical() function

[jdemeyer]

Right now, there is a function radical() as member of IntegerRing_class. But there is no generic radical() function:

sage: radical(100)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)

/usr/local/src/sage-4.4.4/devel/sage-test/<ipython console> in <module>()

NameError: name 'radical' is not defined

[jdemeyer]

The attached patch adds a radical() function, functions Factorization.radical() and Factorization.radical_value(). It also fixes a bug in the radical of a polynomial where the content was not accounted for.

It also changes the output of radical(), in a way which I think makes more sense: radical(0) now gives an error instead of returning 1 and the radical of a negative number is positive (instead of negative).

In order for all doctests to succeed, you need to apply also the patches for #9450.

[kcrisman]

For what it's worth, groups and algebras have radicals too, so it might be worth putting a #TODO in the main definition that one might want radical(G,type='nilpotent') to work here if that ever is implemented/wrapped nicely (e.g. see ​here), so that future developers do not break the possibility of that syntax.

[cremona]

Applies well to 4.6.1.alpha1 and all tests pass.

[jdemeyer]

Thanks, this is something I did in Leiden and totally forgot about.

[cremona]

Replying to jdemeyer:

Thanks, this is something I did in Leiden and totally forgot about.

I cannot see a reason not to include it, despite the comment from kcrisman.