Construction of a Polynomial Having as its Root a Number Generated by Polynomial Roots

If ${\alpha_1...\alpha_n}$ are roots of polynomials ${f_1(x),...,f_n(x)}$ over a field ${K}$ of degrees ${d_1,...,d_n}$ respectively, and ${g(x_1,...,x_n)}$ is a polynomial over ${K}$ with respect to n variables ${x_1,...,x_n}$, then construct a polynomial ${h(x)}$ over ${K}$ which has the root ${g(\alpha_1,...,\alpha_n)}$.

For constructing this, set a polynomial ${H}$ of the form

$\displaystyle H(x_1,...,x_n):= a_Ng(x_1,...,x_n)^N+a_{N-1}g(x_1,...,x_n)^{N-1}+...+a_1g(x_1,...,x_n)+a_0 \ \ \ \ \ (1)$

where ${N=d_1\cdots d_n}$ and the coefficients ${a_0,...,a_N}$ are yet to be determined. Divide ${H}$ by ${f_1(x_1)}$, divide the remainder by ${f_2(x_2)}$ and, iterating this process, finally get a remainder ${r(x_1,...,x_n)}$. All coefficients of ${r}$ can be written as linear combinations of ${a_i}$’s. Any monomial of the polynomial ${r}$ is of the form ${ax_1^{m_1}\cdots x_n^{m_n} (0\leq m_i and thus the polynomial ${r}$ consists of less monomials than ${N+1}$, that is, the number of all coefficients of ${r}$ is less than ${N+1}$.The number of all coefficients of ${H}$ is ${N+1}$. Hence we can choose non-trivial (i.e. some ${a_i}$ is not zero) ${a_i}$’s such that all coefficients of ${r}$ are zero and obtain ${h(x):=a_Nx^N+...+a_0}$, which has the root ${g(\alpha_1,...,\alpha_n)}$.

Example1.

$\displaystyle \begin{array}{rcl} n=4,\alpha_1=\sqrt{2},\alpha_2=\sqrt{3},\alpha_3=\sqrt{5},\alpha_4=\sqrt{7},f_1=x^2-2,f_2=x^2-3,f_3=x^2-5,f_4=x^2-7,g(x_1,x_2,x_3,x_4)=x_1^3+x_2x_3+x_4^5,\\ h(x)=x^8 - 67320 x^6 + 1696399952 x^4 - 18964319566080 x^2 + 79356115963134976 \end{array}$

Example2.

$\displaystyle \begin{array}{rcl} n=2,\alpha_1=\sqrt{-3},\alpha_2=\sqrt[3]{2},f_1=x^2+3,f_2=x^3-2,g(x_1,x_2)=x_1+x_2,\\ h(x)=x^6 + 9 x^4 - 4 x^3 + 27 x^2 + 36 x + 31 \end{array}$