In this book by a leading Russian mathematician and full member of the Russian Academy of Sciences, Igor Rostislavovich Shafarevich, the elements of algebra as a field of contemporary mathematics are laid out based on material bordering the school program as closely as possible.

The book can be used as enrichment materials for students in grades 9-12 in both ordinary schools and schools with a deeper study of mathematics and the sciences and also as a book for mathematics teachers.

Preface

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.Birmmial 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 forr(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

Index