**EDIT2:** After reading some papers, I think the question can best be rephrased as "How can the minimal polynomial for a polynomial with algebraic coefficients be calculated.  I have seen papers and textbooks that show that algebraic numbers are algrebraically closed, but I haven't seen a constructive proof.

> Let $f_n,f_{n-1},...,f_0$ be univariate polynomials with rational
> coefficients.  For each $f_i$, assume that we have successfully
> isolated a root $\lambda_i$ via Sturm's Theorem as the only root
> within the range $[\lambda_i^-,\lambda_i^+]$.
> 
> Define $g$ as the univariate polynomial:
> 
> $$g(x) = \lambda_nx^n + \lambda_{n-1}x^{n-1} + ... + \lambda_0$$
> 
> **Is it possible to isolate the zeros of $g$?  Specifically, is it possible to determine if $g$ has repeated roots?**
> 
> I asked a somewhat similar question [here][1] in which each
> $\lambda_i$ is represented as an interval whose size can be shrunk
> arbitrarily (but not shrunk to a single point).  Alex Degtyarev
> correctly pointed out that the problem cannot be solved if the values
> of $\lambda_i$ are not known exactly.  
> 
> However, in this instance, the values are known exactly. 
> Unfortunately, I'm missing how the rational coefficients of the $f_i$
> can be incorporated in an algorithm to isolate the roots of $g$.
> 
> Thanks for any help.
> 
> **EDIT:**   Since posting the question, I've read a bit on Galois Theory, and it looks like this problem can be solved, although I'm
> still trying to figure out exactly how.  I've figured out algorithms
> to find the minimal polynomial for sums and products of algebraic
> numbers.  I still haven't found a algorithm to determine the minimal
> polynomial for a polynomial with algebraic coefficients although I
> have found a [proof][2] that such a polynomial exists.

  [1]: http://mathoverflow.net/questions/186621/determining-roots-of-a-polynomial-with-interval-estimates-of-coefficients
  [2]: http://planetmath.org/sites/default/files/texpdf/42022.pdf
  [3]: http://math.stackexchange.com/questions/155122/how-to-prove-that-the-sum-and-product-of-two-algebraic-numbers-is-algebraic