Algorithm for checking linear independence of algebraic numbers Is there any if and only if condition for checking $Q$-linear independence of given a set of numbers say $\alpha_i$ ? More precisely how to check linear independence of given $n$ algebraic numbers (algebraic numbers are given by an interval and its irreducible polynomial such that that interval does not contain any other root of that polynomial ) ? Is there any known algorithm for it? 
PS : Def :- $\alpha_i$ are Q- linearly dependent iff $ \exists c_i \in \mathbb{Q} $ such that $\sum c_i\alpha_i=0$   
 A: I had the same question, so I figure I might as well spell it out explicitly for the next person who comes across this post.
For convenience, here is a link to the book WhatsUp mentioned in the comments above. One starts by solving the primitive element problem, discussed on page 181, for one's set of numbers. When testing whether $\alpha_1\in\mathbb{Q}(k_1\alpha_1+\alpha_2)$ using one of the field membership algorithms, on success one gets a polynomial $P_1(x)\in\mathbb{Q}[x]$ such that $\alpha_1=P_1(k_1\alpha_1+\alpha_2)$. Let $\beta_2=k_1\alpha_1+\alpha_2$. Then for $Q_1(x)=x-k_1P_1(x)$, $\alpha_2=Q_1(\beta_2)$. One then similarly finds $P_i(x)$ and $k_i$ such that $\beta_i=P_i(\beta_{i+1})$ where $\beta_{i+1}=k_i\beta_i+\alpha_{i+1}$ and $Q_i(x)$ such that $\alpha_{i+1}=Q_i(\beta_{i+1})$ for each $i$. For $k<i$ one has $\beta_k=P_k(P_{k+1}(...P_{i-1}(P_i(\beta_i))...))$, and substituting this into $Q_{k-1}(x)$, one can get a polynomial for every $a_k$ in terms of the final $b_i$. The matrix whose rows are the coefficients of these polynomials is the relevant one. I don't think it's guaranteed to be square in general, but row reducing and examining the pivots should suffice.
