A book of Abstract Algebra Notes

Chapter 1

Difference between elementary algebra and abstract algebra.

Difference is in the emphasis. Elementary algebra emphasis is on technique. Abstraction is all relative. One person’s abstraction is another person’s bread and butter. The abstract tendency in mathematics is little like the situation of changing moral codes, or changing tastes in music: What shock one generation becomes the norm in the next.

Origins: In Spain barbers were called algebraistas, or bonesetters, because medieval barbers did bonesetting and bloodletting as a sideline to their usual business. Omar Khayyam, who is best remembered for Rubaiyat defined algebra as a science of solving equations. Methods of solving the linear equations and the quadratic were well known even before the Greeks. But nobody had yet found a general solution for cubic equations or quartic (fourth degree) equations .

Cardano’s was  born in 1501 as the illegitimate son of a famous jurist of the city of Pavia. He wrote Ars Magna. In 1535 Tartaglia found a way of solving any cubic equation of the form . Ferrari was a Cardan’s personal servant and discovered a genral method for solving quartic equations of the form . He won fame after defeating Tartaglia in 1548 and received an appointment as supervisor of tax assessments in Mantua. The position bought him wealth and influence. However he couldn’t contain his violent behavior and the story is that he was poisoned by his sister when he was 43 years. Cardano’s son poisoned his wife and he himself was implicated for blasphemy for publishing Christ’s biography. He lost his university position and the rights to publish books and lived on a meagre pension by pope.

Neils Abel showed that there does not exist any formula for the roots of an algebraic equation whose degree is 5 or greater. This sensational discovery brings to a close what is called as the classical age of algebra. Throughout this age algebra was conceived essentially as the science of solving equations.

The modern age