Countability of the algebraic numbers
The set of algebraic numbers is countable: polynomials of degree with integer coefficients inject into , so is a countable union of finite sets.
Theorem 6.10
The set of all algebraic numbers is countable.
Proof
It suffices to show that the polynomials with integer coefficients form a countable set, since is then a countable union of finite sets.
For each , let be the set of degree- polynomials with integer coefficients. The map
injects into , which is countable, so each is countable. A countable union of countable sets is countable, so the polynomials with integer coefficients form a countable set, and hence so does .
Related
Stated in
- Theorem 6.10§6 Countability