Polynomial root of algebraic number

Question

According to Wikipedia, an algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial in one variable with rational coefficients.

The polynomial has many roots, how do I know which of the roots does the algebraic number denotes?

Is this where e.g. "isolating interval representation" comes in, that is, I need to somehow bound which root I'm interested in? What other ways there are to identify a given root?

As you could surely tell, I'm very new to this kind of math...

Answer 1

It follows directly from the definition that every root of a non-zero polynomial with rational coefficients is an algebraic number.

Answer 2

If I've understood them correctly,

1. Each algebraic number has a unique minimal polynomial.
2. More than one algebraic number may have the same minimal polynomial. The set of numbers having the same minimal polynomial are the roots of that polynomial.

One method of identifying the boundaries in the complex plane that contain a given root of a minimal polynomial is the Kollins-Crandick algorithm.

Those rectangles provide a means of uniquely identifying which root of the minimal polynomial you are interested in once the rectangles have been refined enough to be disjoint.

Answer 3

Algebraic numbers that are roots of the same minimal polynomial are indistinguishable algebraically: There is always an automorphism of the field of algebraic numbers that interchanges them. However, these automorphisms are not order automorphisms, therefore you can tell algebraic numbers apart by comparing them using the ordering of the rational numbers, if you can induce an ordering on the algebraic numbers from that. To do this, you don't have to explicitly specify for each possible minimal polynomial the ordering of its roots in a way that consistently turns the algebraic numbers into an ordered superfield of the rationals. You can specify a root using rational numbers to specify an interval, or for complex numbers a circle/rectangle, that contains exactly one root of the minimal polynomial. This of course means you have to be able to tell whether such an interval/rectangle/circle contains a single root, [and there are algorithms for that]. Note that the number of roots in an interval is independent from the concrete ordering of the algebraic numbers.