Let $\varphi_1,\ldots,\varphi_n$ be the roots of a monic polynomial $p$ of degree $n$ with integer coefficients. $\varphi_i$ are thus, by definition, algebraic integers.
Let $a_i$, $i=1,\ldots,n$, be a further set of integers and consider the system of linear equations in $x_i$ given by $$ a_i = \sum_{j=1}^n{x_j \varphi_j^i},\quad i=1,\ldots,n. $$ Let us assume that this system of equations has a unique solution $x_i$.
I seem to recall from my undergraduate training that the $x_i$ are also algebraic numbers, maybe even algebraic integers. I also seem to recall that a bound on their degree is related to the resultant of two polynomials.
Edit: Experimentation suggests that the $x_i$ are algebraic of degree $n$.
Questions
1) Are the $x_i$ algebraic? What is their (maximal) degree?
2) If they are algebraic, as experimentation suggests, how are the coefficients of their minimal polynomials determined by the given data?