Given a system of linear equations in $k$ variables $x_1, x_2, \ldots, x_k$ with coefficients that are complex algebraic numbers of height at most H and degree at most $d$, what is the time complexity of solving this system as a function of $k, d, H$? Is this in polynomial time?
EDIT: Let me fix that the algebraic numbers that are coefficients of the linear equation be represented by the minimal polynomial and the specific root to sufficient accuracy (which is determined by the root separation bound). Does the complexity change if the algebraci numbers are represented as vectors over some extension field $\mathbb{Q}(\alpha)$?