Using various examples this monograph shows that algebra is one of the most beautiful forms of mathematics. In doing so, it explains the basics of algebra, number theory, set theory and probability. The text presupposes very limited knowledge of mathematics, making it an ideal read for anybody new to the subject. The author, I.R. Shafarevich, is well-known across the world as one of the most outstanding mathematicians of this century as well as one of the most respected mathematical writers.

Inhalt

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.