Question

13. An algebraic number is a real number which is the root of a polynomial co + ciz c2n in which all of the coefficients c i 1,2,.,n) are integers. The order of an algebraic number is the smallest natural number n for which z is a root of an n-th degree polynomial with integer coefficients. A real number is transcendental if it is not algebraic. a) Show that the set of algebraic numbers of order n is countable (b) Show that the set of algebraic numbers is countable. c) Show that the set of transcendental numbers is uncountable.

Answer 1

8.9a) The set of polynomials with integer coefficients is countable Proof. First consider the set Pn of polynomials of degree n with nonnegative integer coefficients. First, since there are infinitely many primes, there exists an injective map p N Q which enumerates the set of prime numbers Q C N. Ten there is a map given by By unique factorization, this map is one-to-one (the same argument as the proof that A × B is countable if A and B are countable one could also show that is equinumerous to N × N × N × … × N where there are n factors). Thus Pn is countable. Finally, the set of polynomials P can be expressed as which is a union of countable sets, and hence countable. 8.9b) The set of algebraic numbers is countable. Proof. We know that the set P of polynomials is countable. Each polynomial of degree n has at most n roots, thus for any polynomial p, the set Rp of roots of p is countable. Thus the set A of algebraic numbers can be expressed as is a countable union of countable sets, and hence countable. 8.9c) Since the union of the set of transcendental numbers and the set of algebraic numbers is the set of real numbers, and the set of real numbers is uncountable, we must have that the set of transcendental numbers is uncountable (since the union of two countable sets is countable). Thus there are more transcendental numbers (uncountably many) than algebraic numbers (countably many)

we have proved (b) and (c) ,

Since, A=algebraic number of order n is contained in the set of all algebraic numbers (B), , Since B is countable, A is countable