Suppose we have a set of polynomials $f_1, f_2, \ldots, f_n \in \mathbb{Q}[x]$. Consider the set $$S := \{\alpha \in \mathbb{C} \; | \; f_i(\alpha) = 0 \text{ for some } i \}$$ of complex roots of these polynomials. Then $S$ is a finite set and consequently $V := \operatorname{span}_\mathbb{Q} S$ is a finite-dimensional vector space. (This sort of situation comes up for instance when performing calculations in representation or invariant theory with a computer).
Often such a list will come out of a computer in a form which is sub-optimal -- for instance, it might talk about the numbers such that $\alpha^2 + \alpha + 1 = 0$ and the numbers such that $\beta^2 - \beta + 1 = 0$, when it would be much nicer simply to refer to the numbers $\alpha^2 + \alpha + 1 = 0$ and then the numbers one more than those.
Is there a nice algorithm for calculating $\dim_\mathbb{Q} V$, or for reducing the number of $f_i$ in such a list? Better yet, is there a built-in command to do this in any software?