SUMMARY
The set of all algebraic numbers is proven to be countable by demonstrating that for every natural number N, there are only finitely many polynomials with integer coefficients of degree N. This implies that the roots of these polynomials, which are the algebraic numbers, can be listed in a sequence. Consequently, since each polynomial contributes a finite number of roots, the overall set of algebraic numbers is countable.
PREREQUISITES
- Understanding of polynomial equations and their roots
- Familiarity with the concept of countability in set theory
- Basic knowledge of complex numbers
- Knowledge of integer coefficients in polynomials
NEXT STEPS
- Study the concept of countable vs. uncountable sets in set theory
- Learn about polynomial functions and their properties
- Explore the implications of algebraic numbers in number theory
- Investigate the relationship between algebraic numbers and transcendental numbers
USEFUL FOR
Mathematicians, students studying algebra and number theory, and anyone interested in the properties of algebraic and transcendental numbers.