Let n N Let An be the set of all squareroots of polynomials

Let n N. Let A_n be the set of all squareroots of polynomials of degree n with rational coefficients. Prove that A_n is countable. You may assume the fact that a polynomial of degree n has at most n squareroots. Prove that the set of algebraic numbers is countably infinite.

Solution

Pn is the polynomial of degree n.

Since a polynomial consists of only positive integral powers of n,

the polynomial will have n roots at most.

Since there can be maximum n roots, rational roots will be in number <=n.

Since the roots are finite and <=n, we can find a one to one correspondence with natural numbers.

i.e. if x1, x2,...xm are the rational roots f(x1) = 1, f(xi) =i is the one to one correspondence with natural numbers

Or the set An is countable.