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SMA_01
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Thank you for your help (and patience), I know it took a while for me to get it.
Proving algebraic numbers are countable?
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SUMMARY
The discussion centers on proving that the set of algebraic numbers, denoted as An, is countable by demonstrating that there are only countably many polynomials of degree n with integer coefficients. Participants emphasize using the hint regarding the condition |ai| < m to establish an upper bound on the number of such polynomials. The conclusion is that since each polynomial has a finite number of roots, and the union of countable sets is countable, An is indeed countable.
PREREQUISITES- Understanding of algebraic numbers and their definitions.
- Familiarity with polynomial equations and integer coefficients.
- Knowledge of countable sets and properties of unions of sets.
- Basic principles of mathematical induction.
- Study the concept of countable sets in set theory.
- Learn about polynomial functions and their roots in algebra.
- Explore the proof techniques involving induction and direct proof methods.
- Investigate the properties of unions of countable sets in mathematical analysis.
Mathematics students, educators, and researchers interested in algebra, set theory, and the foundations of number theory.
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