Algorithm for finding irreducible polynomials in finite field extension

Question

Let $K(\alpha_1,\ldots,\alpha_n)/K$ be a finite field extension and suppose we know ${\rm irr}(\alpha_1,K),\ldots,{\rm irr}(\alpha_n,K)\in K[x]$. My question is: given an arbitrary element $\lambda\in K(\alpha_1,\ldots,\alpha_n)$, is there an algorithm for finding ${\rm irr}(\lambda,K)$ with only the preceding information?