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Beschreibung
One of the landmarks in the history of mathematics is the proof of the nonex- tence of algorithms based solely on radicals and elementary arithmetic operations (addition, subtraction, multiplication, and division) for solutions of general al- braic equations of degrees higher than four. This proof by the French mathema- cian Evariste Galois in the early nineteenth century used the then novel concept of the permutation symmetry of the roots of algebraic equations and led to the invention of group theory, an area of mathematics now nearly two centuries old that has had extensive applications in the physical sciences in recent decades. The radical-based algorithms for solutions of general algebraic equations of degrees 2 (quadratic equations), 3 (cubic equations), and 4 (quartic equations) have been well-known for a number of centuries. The quadratic equation algorithm uses a single square root, the cubic equation algorithm uses a square root inside a cube root, and the quartic equation algorithm combines the cubic and quadratic equation algorithms with no new features. The details of the formulas for these equations of degree d(d = 2,3,4) relate to the properties of the corresponding symmetric groups Sd which are isomorphic to the symmetries of the equilateral triangle for d = 3 and the regular tetrahedron for d ¿ 4.
One of the landmarks in the history of mathematics is the proof of the nonex- tence of algorithms based solely on radicals and elementary arithmetic operations (addition, subtraction, multiplication, and division) for solutions of general al- braic equations of degrees higher than four. This proof by the French mathema- cian Evariste Galois in the early nineteenth century used the then novel concept of the permutation symmetry of the roots of algebraic equations and led to the invention of group theory, an area of mathematics now nearly two centuries old that has had extensive applications in the physical sciences in recent decades. The radical-based algorithms for solutions of general algebraic equations of degrees 2 (quadratic equations), 3 (cubic equations), and 4 (quartic equations) have been well-known for a number of centuries. The quadratic equation algorithm uses a single square root, the cubic equation algorithm uses a square root inside a cube root, and the quartic equation algorithm combines the cubic and quadratic equation algorithms with no new features. The details of the formulas for these equations of degree d(d = 2,3,4) relate to the properties of the corresponding symmetric groups Sd which are isomorphic to the symmetries of the equilateral triangle for d = 3 and the regular tetrahedron for d ¿ 4.
Zusammenfassung
An affordable softcover edition of a classic text
Complete algorithm for roots of the general quintic equation
Key ideas accessible to non-specialists
Indroductory chapter covers group theory and symmetry, Galois theory, Tschirnhausen transformations, and some elementary properties of an elliptic function
Discussion of algorithms for roots of general equation of degrees higher than five
Includes supplementary material: [...]
Inhaltsverzeichnis
Group Theory and Symmetry.- The Symmetry of Equations: Galois Theory and Tschirnhausen Transformations.- Elliptic Functions.- Algebraic Equations Soluble by Radicals.- The Kiepert Algorithm for Roots of the General Quintic Equation.- The Methods of Hermite and Gordan for Solving the General Quintic Equation.- Beyond the Quintic Equation.
Details
Erscheinungsjahr: | 2008 |
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Fachbereich: | Arithmetik & Algebra |
Genre: | Importe, Mathematik |
Rubrik: | Naturwissenschaften & Technik |
Medium: | Taschenbuch |
Inhalt: |
viii
150 S. 16 s/w Illustr. 150 p. 16 illus. |
ISBN-13: | 9780817648367 |
ISBN-10: | 0817648364 |
Sprache: | Englisch |
Herstellernummer: | 12532490 |
Einband: | Kartoniert / Broschiert |
Autor: | King, R. Bruce |
Auflage: | 1st edition 1996. 2nd printing 2008 |
Hersteller: | Birkhäuser Boston |
Verantwortliche Person für die EU: | Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, D-14197 Berlin, juergen.hartmann@springer.com |
Maße: | 235 x 155 x 9 mm |
Von/Mit: | R. Bruce King |
Erscheinungsdatum: | 13.11.2008 |
Gewicht: | 0,254 kg |