I'm talking about a computer algorithm in the realm of symbolic computation. I also need to be able to write the code myself as opposed to using a third-party tool.
One way I thought of was to find any polynomials $A(x)$ and $B(x)$ of which nested radicals $\alpha$ and $\beta$, respectively, are roots. This is easier than finding minimal polynomials, I assume.
Find their GCD $G(x)$ and make sure it isn't $1$.
Then $\frac{G(x)}{x-\alpha} = \frac{G(x)}{x-\beta} \implies \alpha = \beta$.
This is right, right?
The expressions can get huge if this is done naively. Is there a neater way?