Author | Shafarevich, Igor R. author |
---|---|

Title | Discourses on Algebra [electronic resource] / by Igor R. Shafarevich |

Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2003 |

Connect to | http://dx.doi.org/10.1007/978-3-642-56325-6 |

Descript | X, 279 p. 2 illus. online resource |

SUMMARY

I wish that algebra would be the Cinderella ofour story. In the mathยญ ematics program in schools, geometry has often been the favorite daughยญ ter. The amount of geometric knowledge studied in schools is approxยญ imately equal to the level achieved in ancient Greece and summarized by Euclid in his Elements (third century B. C. ). For a long time, geomยญ etry was taught according to Euclid; simplified variants have recently appeared. In spite of all the changes introduced in geometry coursยญ 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 didn't choose math as my profession, but I'll 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 alยญ gebra. Algebra courses in schools comprise a strange mixture of useful rules, logical judgments, and exercises in using aids such as tables of logยญ 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 conยญ tinued from the Renaissance in western Europe. Nevertheless, algebra is just as fundamental, just as deep, and just as beautiful as geometry

CONTENT

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 -- 3. Finite Sets (Topic: Sets) -- 7. Sets and Subsets -- 8. Combinatorics -- 9. Set Algebra -- 10. The Language of Probability -- 4. Prime Numbers (Topic: Numbers) -- 11. The Number of Prime Numbers is Infinite -- 12. Eulerโ{128}{153}s Proof That the Number of Prime Numbers is Infinite -- 13. Distribution of Prime Numbers -- 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 -- 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 -- Dates of Lives of Mathematicians Mentioned in the Text

Mathematics
Algebra
Number theory
Mathematics
Algebra
Number Theory