Numerical Methods for Roots of Polynomials - Part II: Chapter 13. Existence and Solution by Radicals

Numerical Methods for Roots of Polynomials - Part II: Chapter 13. Existence and Solution by Radicals
ISBN-10
0128077034
ISBN-13
9780128077030
Series
Numerical Methods for Roots of Polynomials - Part II
Category
Mathematics
Pages
728
Language
English
Published
2013-07-19
Publisher
Elsevier Inc. Chapters
Authors
J.M. McNamee, V.Y. Pan

Description

We consider proofs that every polynomial has one zero (and hence n) in the complex plane. This was proved by Gauss in 1799, although a flaw in his proof was pointed out and fixed by Ostrowski in 1920, whereas other scientists had previously made unsuccessful attempts. We give details of Gauss’ fourth (trigonometric) proof, and also more modern proofs, such as several based on integration, or on minimization. We also treat the proofs that polynomials of degree 5 or more cannot in general be solved in terms of radicals. We define groups and fields, the set of congruence classes mod p (x), extension fields, algebraic extensions, permutations, the Galois group. We quote the fundamental theorem of Galois theory, the definition of a solvable group, and Galois’ criterion (that a polynomial is solvable by radicals if and only if its group is solvable). We prove that for the group is not solvable. Finally we mention that a particular quintic has Galois group , which is not solvable, and so the quintic cannot be solved by radicals.

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