Vtyjkyuk

Here is my version of the proof that the set of algebraic numbers is countable. Is it correct?

First we show that the set of polynomials with rational coefficients is countable. For ever fixed natural $n$, there is a bijection between the set of polynomials of degree $n$ and the set $mathbb Q^n$ that sends $x^n+a_n-1x^n-1+dots+a_0$ to $(a_n-1,dots,a_0)$. Thus the set of polynomials with rational coefficients of a fixed degree is countable. The set of all polynomials with rational coefficients is the union of that set over all $nin mathbb N$, so it's also countable as a countable union of countable sets.

Now each polynomial of degree $n$ has at most $n$ roots. We can enumerate all polynomials due to the above: $p_1(x),p_2(x),dots.$ Let $(P)_iin mathbb N$ be the set of all polynomials with rational coefficients. Let $S_i$ be the set of all roots of $p_i(x)$. The set of all algebraic numbers is the union of $S_i$ over all $iin mathbb N$. This is a countable union of finite sets, thus countable.