Discourses on Algebra

1. Integers (Topic: Numbers).- 1. ?2 Is Not Rational.- 2. The Irrationality of Other Square Roots.- 3. Decomposition into Prime Factors.- 2. Simplest Properties of Polynomials (Topic: Polynomials).- 4. Roots and the Divisibility of Polynomials.- 5. Multiple Roots and the Derivative.- 6. Binomial Formula.- Supplement: Polynomials and Bernoulli Numbers.- 3. Finite Sets (Topic: Sets).- 7. Sets and Subsets.- 8. Combinatorics.- 9. Set Algebra.- 10. The Language of Probability.- Supplement: The Chebyshev Inequality.- 4. Prime Numbers (Topic: Numbers).- 11. The Number of Prime Numbers is Infinite.- 12. Euler’s Proof That the Number of Prime Numbers is Infinite.- 13. Distribution of Prime Numbers.- Supplement: The Chebyshev Inequality for ?(n).- 5. Real Numbers and Polynomials (Topic: Numbers and Polynomials).- 14. Axioms of the Real Numbers.- 15. Limits and Infinite Sums.- 16. Representation of Real Numbers as Decimal Fractions.- 17. Real Roots of Polynomials.- Supplement: Sturm’s Theorem.- 6. Infinite Sets (Topic: Sets).- 18. Equipotence.- 19. Continuum.- 20. Thin Sets.- Supplement: Normal Numbers.- 7. Power Series (Topic: Polynomials).- 21. Polynomials as Generating Functions.- 22. Power Series.- 23. Partitio Numerorum.- Supplement 1: The Euler Pentagon Theorem.- Supplement 2: Generating Function for Bernoulli Numbers.- Dates of Lives of Mathematicians Mentioned in the Text.