Algebraic Numbers Countable

Algebraic number can be defined as Set of numbers which is countable. In a simple way, algebraic number can be expressed as roots of non-zero Polynomials. For instance, all Rational Numbers are algebraic number if they follow a condition, that p and q are two Integer numbers and q ≠ 0 then rational numbers will be called as Algebraic Numbers since (y = p / q)  is the root of  (q y – p). Consider a quadratic surd (irrational root of a quadratic polynomial) which has integer coefficients P y² + Q y + R. Here P, Q and R are integer coefficients and known as algebraic numbers.

Let’s see algebraic numbers countable by following example.

Set below can be said as algebraic number if all integers P_ 0... P_ t are not zero. That is,
P_ 0 X ^t + P_ 1 X ^t-1 +...+ P_ ^t-1 X + P_ 0 = 0.
 
If we want to prove that set of all algebraic numbers is countable, following procedure is used. Consider a set P_ 1 of algebraic numbers in such a way that P_ 0 X + P_ 1 = 0. It simply expresses the rational numbers, and therefore is countable. Let’s suppose that set P_ n, of algebraic numbers P_ 0 X t +...+ P_ n = 0, is countable. Then again suppose that set P_ n + 1, of algebraic numbers P_ 0 X ^{t + 1} + P_ 1 X ^t +...+ P_ ^{t + 1} = 0. Now select a fixed P_ 0 = P, then this subset of P_n+1 can be put in One-to-One Correspondence with P_n and therefore is  countable.