Indexing of Roots of Polynomials

The indexing of roots of a polynomial (sometimes written as \(\text{Root}[p,i]\)) takes the real roots first, in increasing order. For polynomials with rational coefficients, the complex conjugate pairs of roots have consecutive indices where the roots with a negative imaginary part have a lower index than their conjugate. The following picture shows a plot of the polynomial \(p = (x-1) \left(x^2-3\right) \left(x^2+x+2\right) \left(x^3+1\right)\) and the indexing of its roots.

If a polynomial has non-rational coefficients then its roots might be described via expressions of the form \(\text{Root}[\{p_1,\ldots,p_n\},\{k_1,\ldots,k_n\}]\) where the \(p_i\) form a triangular system of polynomials and the value of this root is defined as follows: Given equations \(p_1(x_1) = 0,\ p_2(x_1,x_2) = 0, \ldots, p_n(x_1,\ldots,x_n) = 0\) we recursively define \(r_1\) as the \(k_1'\)th root of \(p_1(x_1) = 0\), \(r_2\) as the \(k_2'\)th root of \(p_2(r_1,x_2)=0\), and finally \(r_n\) as the \(k_n'\)th root of \(p_n(r_1,\ldots,r_{n-1},x_n)=0\). The represented root is then the value of $r_n$