代数讲义

Preface

1. Integers （Topic： Numbers）

1.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. Eulers 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： Sturms 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.

I wish that algebra would be the Cinderella of our story. In the math-ematics program in schools, geometry has often been the favorite daugh-ter. The amount of geometric knowledge studied in schools is approx-imately equal to the level achieved in ancient Greece and summarized by Euclid in his Elements （third century B.C.）. For a long time, geom- etry was taught according to Euclid; simplified variants have recently appeared. In spite of all the changes introduced in geometry cours-
es, geometry retains the influence of Euclid and the inclination of the grandiose scientific revolution that occurred in Greece. More than once I have met a person who said, "I didnt choose math as my profession,but Ill never forget the beauty of the elegant edifice built in geometry with its strict deduction of more and more complicated propositions, all beginning from the very simplest, most obvious statements！"
Unfortunately, I have never heard a similar assessment concerning al-gebra. Algebra courses in schools comprise a strange mixture of useful rules, logical judgments, and exercises in using aids such as tables of log-arithms and pocket calculators. Such a course is closer in spirit to the brand of mathematics developed in ancient Egypt and Babylon than to the line of development that appeared in ancient Greece and then con-tinued from the Renaissance in western Europe. Nevertheless, algebra is just as fundamental, just as deep, and just as beautiful as geometry.Moreover, from the standpoint of the modern division of mathemat-ics into branches, the algebra courses in schools include elements from several branches： algebra, number theory, combinatorics, and a bit of probability theory.