Exercise 2.2

Let $𝔸$ be the set of algebraic numbers. Let $p$ be a polynomial over $ℂ$. Using the division algorithm for polynomials over a field, we observe that $(z-\alpha )\mid p(x)$ iff $\alpha$ is a root of $p$. We can deduce that a polynomial of degree $n$ has at most $n$ roots.

Let $P_n$ be the set of polynomials $p$ in $\mathbb{Z}[x]$ for which the coefficients $a_i$ satisfy ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^m|a_i|=n-\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p$. This is a finite set, and $\bigcup <!--\; FUNCTION\; APPLICATION\; -->P_n=\mathbb{Z}[x]$. Let $V_n$ be the corresponding set of all roots of the polynomials in $P_n$. By the above observation, this set is finite. Consequently, $𝔸={\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_n{V}_n$ is at most countable. Since $\mathbb{Z}\subset 𝔸$, it is also at least countable.