@BalarkaSen: I've always assumed it meant some answer expressible with a $\sqrt[N] {\dots}$ but upon some quick research, that wasn't at all what it meant
@Nick Here's something for you to think : Can you give me something which cannot be expressed by radicals, i.e., not a "radical expression"?
Note that radical expressions consists of rational numbers, their addition, multiplication, division, subtraction and their roots (and composition of all of those).
Think about it. It's not hard, just takes a little bit of thinking.
Note that radical expressions consists of rational numbers, their addition, multiplication, division, subtraction and their roots (and composition of all of those).
Roots of polynomial equations with coefficients being rational.
I am talking about algebraic numbers over rationals.
OK, I am going to go for a few hours or so until you digest that. When you do, here's another question for you : Every radical is an algebraic but is every algebraic a radical?
--___-- i've spent the past 2 weeks trying to open that in my browser. It didn't work last thursday, last saturday or even today morning. It's not going to open now.
@Nick more fun : add a point $w_2 = -\sqrt{z}$ in the complex plane. naturally, $w_1$ and $w_2$ would sit on the opposite side of the unit sphere around them :
So, rather, $w_1$ and $w_2$ are "exchanged" while $z$ moves from the +- coordinate to -- coordinate.
@Nick I have just taught you fundamentals of geometric galois actions. :D
I'll let you digest what I said above, perhaps trying it by your own hands in geogebra would be better when you get time. Think about it, and let me know later if you want to see more of this fun stuff.
@Nick just to make you a bit more satisfied, note that continuity of $\sqrt{z}$ is preserved if you define $\text{Sqrt}(z)$ to be the piece-wise function which is $\sqrt{z}$ while $z$ is on the +1 coordinate, and $-\sqrt{z}$ when in the other coordinate.
Now take two sheets of papers, one called "$\sqrt{z}$" and one called "-\sqrt{z}".
By the construction above we have taken to preserve continuity, the two sheets are cut along the negative real axis and pasted criss-crossed.
Just to let you know, I have shown you very advanced stuff found by V. I. Arnold formalized by world-famous mathematician Grothendeick in elementary high-school mathematics. Of course, this is just a scratch on the surface.
