Algorithm for computing algebraic numbers? (Why are algebraic numbers computable?) Suppose $b$ is algebraic over the rationals.
In other words:
$p(b) = 0$ for some polynomial where all the coefficients are rational.
I am told $b$ is computable. But why?


*

*Can I derive a polynomial from $p$ that I can evaluate to get $b$?
edit: commentor clearly pointed out no. Rationals are closed under those operations.

*Is there some other algorithm for computing $b$ given the polynomial? edit: accepted answer: a root finding algorithm.

*If not what is the general argument that algebraic numbers are computable? edit: an argument that the root finding algorithm will converge.

The answer is that iterative root-finding algorithms, like the Aberth method, exist to numerically find the roots of polynomials whose coefficients are themselves computable.  Therefore if p(x) has rational coefficients, there exists a program that will produce b.  So b is computable.
