need help explaining the complex roots of a cubic

Question

I am trying to understand a Galois theory example and we are looking at the solutions of $x^3-2=0$. It says they are $2^\frac{1}{3},2^\frac{1}{3}\omega, \text{ and } 2^\frac{1}{3}\omega^2$. I know the roots are $\sqrt[3]{2}, -\sqrt[3]{-2}$ and $(-1)^\frac{2}{3}\sqrt[3]{2}$. How do we get this $\omega$ notation?

Answer 1

basically, here $w$ is a third root of unity , though it is multiplied by $2$ to account for the modulus. One reason you use the notation is that the roots of unity are a cyclic group , generated by $w$, so that the roots are $w, w^2, w^3$ (Again, scaled by the modulus which is here $2^{1/3}$). We have $$z^3=e^{i3\theta}=1=e^{i(2\pi+2k\pi)};k=0,1,2 $$, so that the roots are a multiplicative cyclic group generated by $w=e^{i2\pi/3}$, and these are $2^{1/3}w, 2^{1/3}w^2, 2^{1/3}w^3=2^{1/3}$ (since $w^3=1$). This is one way of understanding this notation. (note that this is a cyclic group only for roots of unity.)

Answer 2

The $\omega$ notation is the primitive root. We know that $i$ is the square root of -1, but $-i$ also fits that description. Consider $i^2=-1=(-i)^2$. So we want to do something similar with third roots of 2. So we need to choose whether we want to talk about the analog to i (omega) or the analog to negative i. Because we want to differentiate between these roots, even if there is no reason apriori that we can (because they have the same properties in the field structure), we can introduce this notation. If you are reading from Dummit and Foote, I believe they use a zeta.

Answer 3

The notation you give for the roots is ambigous. The point to realise is that there are three cube roots of any number just as there are two square roots of any number. And just as the two square roots differ by $-1$, the square root of $1$, so do the cube roots differ by a cube roots of one, you can calculate this but the $\omega$ is a shorthand notation for a cube root of $1$. Notice that for many (but not all) applications and arguments in Galois theory the three roots are indistinguishable.