Suppose one has a set of numbers $S$, each of which is algebraic over a field $F$. Does knowing the minimal polynomial over $F$ of each element of $S$ help to determine whether $S$ is linearly independent over $F$?
The context is the case of a number $N=\sum_{i=1}^{n}a_i$ where I already know the minimal polynomial of each $a_i$, in a program I'm working on for calculating minimal polynomials. Let $d_i$ be the degree of the minimal polynomial of $a_i$, and $D=\prod_{i=1}^{n}d_i-1$. I start by calculating powers $1$ through $D$ of $N$ in its sum form, treating it as a multivariate polynomial with each $a_i$ a separate variable, and using the minimal polynomials of the $a_i$ to rewrite any terms whose degree with respect to $a_i$ is greater than or equal to $d_i$ as expressions of lower degree. This gives me $D$ polynomials with at most $D$ terms each, with the $k$th polynomial representing $N^k$. I treat the coefficients of the polynomials as a matrix, the $N^k$ its augmentation, and triangularize the matrix to find a linear combination of the $N^k$ that equals a constant. Subtracting the constant from the linear combination gives an annulling polynomial of $N$. I then factor this polynomial over the rationals and, for each of its factors, replace each $x^k$ with $N^k$ in its sum form in terms of the $a_i$ in order to see whether the factor is still annulling. I assume this last step can fail to detect that a factor is annulling in cases where the $a_i$ are linearly dependent, but detecting whether they are in general seems like a hard problem.