Hello everyone! Once again there are no answers in the book so this is just another check just to make sure. Thank you very much in advance.

Question: "Are the set of all algaebraic numbers countable? Justify your respose"

I will attempt to prove that the algaebraic numbers are countable. Once again for my sake let

be the set of all algaebraic numbers.

Answer: Let

be an algaebraic number. Then by definition there exists a polynomial

, such that

. Therefore let us define the number

by the n-tuple

. So now that we have shown that each algaebraic number may be expressed as a n-tuple we may state that

where

. So now since we have shown that the set of all algaebraic numbers may expressed as a set of n-tuples with each element of the n-tuple being an element of the integers, a countable set, we may conclude that the algaebraic numbers are countable

I know its relatively simple, but I just want to sure I am doing this right.