MATH 414
SPRING 2004
Supplementary Homework Problem
1. A number x ∈ R is said to be algebraic of degree n if it is a root
of a n’th degree polynomial P (x) = an xn + an−1 xn−1 + . . . a1 x + a0
where a0 , a1 . . . an ∈ Z. The number x is algebraic if it is algebraic of
some finite degree n and is transcendental if it is not algebraic.
a) Show that if m ∈ N and q ∈ Q then x = mq is algebraic.
b) Show that the set of algebraic numbers of degree n is countable.
(Suggestions: Find a one to one correspondence between the set of all
n’th degree polynomials with integer coefficients and a subset of the
Cartesian product Zn+1 . You may use the fact that any n’th degree
polynomial has at most n real roots.)
c) Show that the set of all algebraic numbers is countable.
d) Show that the set of transcendental numbers is uncountable.