Chapter 1.7, Problem 4E

(a) A real number $x$ is said to be algebraic (over the rationals) if it satisfies some polynomial equation of positive degree

$x^n+a_{n-1}{x}^{n-1}+\cdot \cdot \cdot +a_{1}x+a_0=0$

with rational coefficients $a_i$. Assuming that each polynomial equation has only finitely many roots, show that the set of algebraic numbers is countable.

(b) A real number is said to be transcendental if it is not algebraic. Assuming the reals are uncountable, show that the transcendental numbers are uncountable. (It is a somewhat surprising fact that only two transcendental numbers are familiar to us: $e$ and $\pi$. Even proving these two numbers transcendental is highly nontrivial.)