# Recitation 17

In section D, I proved that the set of all algebraic numbers is countable.

An algebraic number is a real number that is one of the roots of a polynomial with rational coefficients. For example, all rational are algebraic. Also, the square root of a rational is algebraic.

Let $P$ denote the set of all polynomials with rational coefficients and $P_n$ be the set of all such polynomials with degree $\leq n$. Because there is a bijection between $P_n$ and the cartesian product of $n$ many $\mathbb{Q}$‘s, $P_n$ is countable.

Since $P=\bigcup_{n=0}^\infty P_n$, $P$ is countable as well. Notice that each polynomial has only finte roots. As $P$ is countable, the roots generated by the polynomials in $P$ are countable.