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 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 that such a polynomial exists.

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 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 that such a polynomial exists.

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 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 that such a polynomial exists.

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 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 that such a polynomial exists.

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 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.

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 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 that such a polynomial exists.