bug in simplify_radical

[zimmerma]

the documentation of simplify_radical says:

sage: x.simplify_radical?
...
       Simplifies this symbolic expression, which can contain logs,
       exponentials, and radicals, by converting it into a form which is
       canonical over a large class of expressions and a given ordering of
       variables

however if indeed it is able to recognize zero:

sage: a=1/(sqrt(5)+sqrt(2))-(sqrt(5)-sqrt(2))/3
sage: a.simplify_radical()
0

it does *not* return a canonical expression:

sage: a1=1/(sqrt(5)+sqrt(2))
sage: a2=(sqrt(5)-sqrt(2))/3
sage: a1.simplify_radical()
1/(sqrt(2) + sqrt(5))
sage: a2.simplify_radical()
-1/3*sqrt(2) + 1/3*sqrt(5)
sage: (a1-a2).simplify_radical()
0

[zimmerma]

Note the original question posed to me was: how to multiply say 1/(1+sqrt(2)+sqrt(3)) by the conjugate expression?

[mhansen]

This is the full docstring from Maxima:

Simplifies expr, which can contain logs, exponentials, and radicals, by converting it into a form which is canonical over a large class of expressions and a given ordering of variables; that is, all functionally equivalent forms are mapped into a unique form. For a somewhat larger class of expressions, radcan produces a regular form. Two equivalent expressions in this class do not necessarily have the same appearance, but their difference can be simplified by radcan to zero.

    For some expressions radcan is quite time consuming. This is the cost of exploring certain relationships among the components of the expression for simplifications based on factoring and partial-fraction expansions of exponents. 

Perhaps we should include this

[zimmerma]

Perhaps we should include this

yes (unless of course upstream finds a way to get a real canonical form). And maybe adding an example showing the difference when checking for 0.