Algebraic Numbers Countable Proof

Algebraic number is that number which is obtained from roots after solving a polynomial expression or a Quadratic Equation. Algebraic number can be expressed as roots of non zero Polynomials. All algebraic Numbers can be considered as computational numbers and hence they can be defined. Lets us see following instances where numbers √5 can be considered as algebraic number since it is obtained from roots of polynomial y² – 5. Also, the root of polynomial (y² – y – 1) is known as golden Ratio and it is also an algebraic number. In a view, all Algebraic Numbers are countable that is it’s possible to count them. Let’s see algebraic numbers countable proof. Let’s consider A (y) as algebraic function such that:
A (y) = 0,
 
Here A (y) is a polynomial expression with Integer coefficient P n and P 0 is greater than zero. Polynomial A (y) is an irreducible expression. For each integer M = 2, 3, 4… equation A (y) = 0, exists such that n + P 0 + │P ₁│+…. +│P _n│= M.
For example, if M = 3, then one has the equations y - 1 = 0 and y + 1 = 0 and
 
 
Hence only a fix Set of algebraic numbers will be called as the roots of these equations. These algebraic numbers may be ordered to a sequence FM by the use of an ordered system. For instance, by magnitude of the real part and the imaginary part, concatenated sequences are formed as
F ₂,F ₃, F ₄,.....
Above equation is representing the counting nature of algebraic numbers.