The History of Mathematics: A Brief Course

442 15. MODERN ALGEBRA where a^3 = 1, á ø 1. He argued that since the original equation was symmetric in a, b, and c, the resolv ...

THEORY OF EQUATIONS 443 The technique of counting the number of different values the root of the re- solvent will have when th ...

444 15. MODERN ALGEBRA 1.6. Ruffini. As it turned out, Gauss had no need to publish his own research on the quintic equation. In ...

THEORY OF EQUATIONS 445 Ruffini had shown that it was not possible to exhibit a function of five variables that could be chang ...

446 15. MODERN ALGEBRA 1.8. Abel. Cauchy's work had a profound influence on two young geniuses whose lives were destined to be v ...

THEORY OF EQUATIONS 447 of mathematics Leonard Eugene Dickson as "a very complicated reconstruction of Abel's proof." Hamilton ...

448 15. MODERN ALGEBRA we associate with a group: the composition of permutations. However, it is clear from his language that w ...

THEORY OF EQUATIONS 449 all concrete objects—permutations of the roots of equations. He developed Galois theory to the extent ...

450 15. MODERN ALGEBRA As elliptic integrals—integrals containing the square root of a cubic or quartic polynomial—became better ...

ALGEBRAIC STRUCTURES 451 2. Algebraic structures The concept of a group was the first of the many abstractions that make up th ...

452 15. MODERN ALGEBRA in the Journal fur die reine und angewandte Mathernatik in 1914. In his introduc- tion Fraenkei cited the ...

ALGEBRAIC STRUCTURES 453 likely it was his physical intuition that led him to make this discovery. A rotation in two-dimension ...

454 15. MODERN ALGEBRA the structure constants that constitute the multiplication table for the elements of a basis. Even so, th ...

ALGEBRAIC STRUCTURES 455 under an infinite group of translations of the plane. Klein introduced the concept of an automorphic ...

(^456) 15. MODERN ALGEBRA Since we have no need to do any computations, we omit the other two coordinates. The point to be notic ...

ALGEBRAIC STRUCTURES 457 defines. Essentially, this representation was introduced by the American logician philosopher Charles ...

(^458) 15. MODERN ALGEBRA 1980s with the discovery of the last "sporadic" group. The project consists of so many complicated par ...

QUESTIONS AND PROBLEMS 459 Ideal complex numbers. Numbers of the form m + çù, where ù = -1/2 + v^-3/2, a primitive cube root of ...

460 15. MODERN ALGEBRA 15.3. Cauchy's theorem that every cycle of order 3 can be written as the compo- sition of two cycles of o ...

Part 6. Analysis The History of Mathematics: A Brief Course, Second Edition by Roger Cooke Copyright © 200 5 John Wiley & So ...