A real number is said to be algebraic if it is a root of a polynomial equation a n x n +….+ a 1 x + a 0 =0 with integer coefficients. Note that the algebraic numbers include the rationals and all...

A real number is said to be algebraic if it is a root of a polynomial equation

a_nxⁿ
+….+ a₁x + a₀=0

with integer coefficients. Note that the algebraic numbers include the rationals and all roots of rationals (such as
,
, etc). If a number is

not algebraic, it is called transcendental.

(a) Show that the set of polynomials with integer coefficients is countable.

(b) Show that the set of algebraic numbers is countable.

(c) Are there more algebraic numbers or transcendental numbers?