1. The problem statement, all variables and given/known data I'm working through a problem in Abott's Understanding Analysis, second edition, the statement of the problem being: "Fix a member n of the natural numbers and let An be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n. Using the fact that every polynomial has a finite number of roots, show that An is countable." 2. Relevant equations None given, but possibly helpful are: Theorem (given as 1.5.8): The union of a possibly infinite collection of countable sets is countable Definition: A set is countable if a bijection exists between the given set and the natural numbers. 3. The attempt at a solution I've been struggling endlessly on this problem but I think I'm getting somewhere. My approach is that I first want to show that each configuration of n coefficients can be represented by an ordered n-tuple, and that each such n-tuple is related bijectively to only one set of roots. Then I want to show that the set of coefficient n-tuples is countable, and that's where I'm really stuck, I get the gist of it, that since each coefficient is an integer and therefore a member of a countable set, and there are as many possible coefficient n-tuples as there are n * the cardinality of Z, (which would therefore be countable) but I'm not sure how to make that into a proof. Could I use the theorem that we proved in an example problem in the chapter that for a collection of countable sets, their union is countable? Anyway, what I'm hoping that would result in is a bijection between N and the coefficient n-tuples and a bijection between the coefficient n-tuples and the roots of their corresponding polynomials (ie the algebraic numbers) and therefore composition of these two bijections would result in a function relating N to the algebraic numbers. Or am I just going about this completely the wrong way?