It is well known that not all roots or zeroes of polynomials degree $5$ or higher cannot be expressed in radical forms and must be approximated. Let $Q$ be such a polynomial and $σ$ be a root of $Q$. Let $d$ be the degree of $Q$. If $P$ is a degree $d_2$ polynomial that is a composite function of $Q$ and $d | d_2$. Is it possible to express the roots of $P$ in terms of $σ$ if $(d_2/d)<5$?
For example, let $Q=x^5 + 2 x^3 + x^2 + 1$ be one of these polynomials and $σ$ be a root of $Q$. Let $P = x^{10} + 4 x^8 + 2 x^7 + 4 x^6 + 6 x^5 + x^4 + 4 x^3 + 2 x^2 + 2$. Is it possible to express the roots of $P$ in terms of $σ$ given that $P = Q^2 + 1$?
I would expect something similar to that of $x$ and $x^2+1$: The roots of $x$ are $0$, and the roots of $x^2+1$ are $±i$. $\sqrt (0-1) = ±i$. Maybe something similar could be apply in the example I gave?
Thanks for help in advance.
Here is a particular example of a degree twelve polynomial I constructed using roots from a cubic polynomial:
$Q=x^3 - 7 x - 7$. One root of $Q$ is given by:
$σ = -(7^{2/3}*(1 + i\sqrt(3)))/(2^{2/3} (3*(9 + i\sqrt(3)))^{1/3}) - ((1 - i\sqrt(3)) ((7/2)*(9 + i\sqrt(3)))^{1/3})/(2*3^{2/3})$
(note that $σ$ may not be expressible in radicals sometimes)
If $P=x^{12} - 5 x^8 - 14 x^6 + 76 x^4 + 224 x^2 + 349$, then one root of $P$ is given by:
$\sqrt((σ-i\sqrt(3)))$.