Ugly representation of square roots in console mode

Dear Alasdair, and all !

you write you use root.
What roots do you use : from AlgebraicNumber, Expression or RealClosure.

Can you explain how you understand theses roots ?

I feel (but I can't be sure) that roots in algebraicNumber can be seen 
as the FractionField over polynoms, with some bugs and inconsistencies.

realClosure only handles real roots, and positive root for even radix.

RCFI => RealClosureFractionInteger
(-2::RCFI)^(1/3) + (2::RCFI)^(1/2) --- is zero.

But there are also surprising results :
(4::RCFI)^(1/3) + (2::RFCI)^(2/3)
I can't get the numeric value, and this domain can't be use in 
Expression Integer.

You find some don't so funny results in :
http://axiom-wiki.newsynthesis.org/285EqualInAlgebraicNumberFails

Other CAS compute over usual complex roots with the first positive argument.
(-1)^(1/2) = e^(%i*%pi)^(1/2) = e^(%i*%pi/2) = %i.

What roots do you use with your students ? What point of view do you 
explain ?

The only sens my students have about root is the real root,
they learn the complex ones, but I can't expose AlgebraicNumber.

The exercice I like to solve with axiom :

P(z)=z^4 + 2*m*z^2 + 1, with m real.

solve P(z)=0, (with 4 radix expressions)
find limits for the 4 expressions when m is near of +/-1 or +/-%infinity
and plot with 4 colors the 4 roots. You get a circle and the 2 axes.

You may change z^4 and z^2 by z^6 and z^3, and +1 by -1, for other 
pretty plots.

François



Have a nice day !

F.