12. A real number x is called algebraic if there is a positive integer n and integers aŋ, a1, 7 0. Let A be the set of all algebraic numbers. ..., an such that a, + a¡ x + a, x² +· · . +aj x + a2 x +·...

Extracted text: 12. A real number x is called algebraic if there is a positive integer n and integers aŋ, a1, 7 0. Let A be the set of all algebraic numbers. ..., an such that a, + a¡ x + a, x² +· · . +aj x + a2 x +· • · +a, x" = 0 and an (a) Prove that QC A. (b) Prove that /2 e A. (c) Prove that A is countable. Note: You may use the fact that if n is a positive integer, ɑŋ, ɑ1, ..., an are integers, and a, + 0, then {x ER | do + a1 x + a2 x2 +. +a, x" = 0} is finite. ..