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<d_i, i=1,...,n, a \in K)} 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)}.


\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}


\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}