Let $t=a+b\sqrt{n}\in \mathbb{Q}(\sqrt{n})$ be of norm $1$, i.e $t \cdot\bar{t}=1$ (for instance, $\sqrt{2}-1, \frac{\sqrt{5}-1}{2}$).
Then let $\alpha=\sqrt[p]{t}-\sqrt[p]{t^2}-\sqrt[p]{t^{-1}}-\sqrt[p]{t^{-2}}$, where $p$ is an arbitrary prime. (Similar results hold if some of the signs in the sum are flipped, say $\beta=\sqrt[p]{t}-\sqrt[p]{t^2}+\sqrt[p]{t^{-1}}+\sqrt[p]{t^{-2}}$)
Then after some testing $\alpha$ seems to have some amazing properties, but I am not sure how to prove or disprove them:
First, the minimal polynomial $m(x)$ of $\alpha$ is of degree $p$. I guess this would mean $m(x)$ is irreducible and solvable, as the Galois group has an element of order $p$.
Second, it seems that if $p$ is large enough, we have an explicit expression for the first several terms of $m(x)$: $$m(x)=x^p+2p \cdot x^{p-3}+3p \cdot x^{p-4}+4p\cdot x^{p-5}+...$$ For example, the minimal polynomial of $$\alpha=\sqrt[11]{\sqrt{2}-1}-\sqrt[11]{\sqrt{2}+1}-\sqrt[11]{3-2\sqrt{2}}-\sqrt[11]{3+2\sqrt{2}}$$ is $$m(x)=x^{11}+22 x^8+33 x^7+44 x^6+165 x^5+484 x^4+781 x^3+726 x^2+1001 x+1108$$
I am not sure if these results are evident but I do not know how to tackle them. Is it possible to prove the results without doing heavy calculations? Any help would be appreciated.