Algebraic number

The golden ratio (φ) is an algebraic number since it is a solution of the quadratic equation x²-x-1=0, whose coefficients are integers.

The square root of a rational number, √, is an algebraic number since it is a solution of the quadratic equation nx²-m=0, whose coefficients are integers.

Thus, the equation x-eʸ#61;0 is satisfied for x#61;1,#92;y#61;0 and for no other pair of algebraic numbers.

As a matter of fact, it is not true that the algebraic numbers above can be factored uniquely, but the first case of failure occurs when p = 23.

(i) It is easily checked that the set of all algebraic numbers is countable, whereas the set of all complex numbers is uncountable (this non-constructive proof goes back to Cantor).