Browse > Article
http://dx.doi.org/10.14477/jhm.2014.27.3.165

Lagrange and Polynomial Equations  

Koh, Youngmee (Dept. of Math., The Univ. of Suwon)
Ree, Sangwook (Dept. of Math., The Univ. of Suwon)
Publication Information
Journal for History of Mathematics / v.27, no.3, 2014 , pp. 165-182 More about this Journal
Abstract
After algebraic expressions for the roots of 3rd and 4th degree polynomial equations were given in the mid 16th century, seeking such a formula for the 5th and greater degree equations had been one main problem for algebraists for almost 200 years. Lagrange made careful and thorough investigation of various solving methods for equations with the purpose of finding a principle which could be applicable to general equations. In the process of doing this, he found a relation between the roots of the original equation and its auxiliary equation using permutations of the roots. Lagrange's ingenious idea of using permutations of roots of the original equation is regarded as the key factor of the Abel's proof of unsolvability by radicals of general 5th degree equations and of Galois' theory as well. This paper intends to examine Lagrange's contribution in the theory of polynomial equations, providing a detailed analysis of various solving methods of Lagrange and others before him.
Keywords
Lagrange; polynomial equation; algebraically solvable; solvable by radicals; auxiliary equation; reslovent;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Rider R. Hamburg, The theory of equations in the 18th century: The work of Joseph Lagrange, Archive for History of Exact Sciences 16 (1976/77), 17-36.
2 Melvin B. Kiernan, The development of Galois theory from Lagrange to Artin, Archive for History of Exact Sciences 8 (1971), 40-154.   DOI
3 Ng Tuen Wai, History of solving polynomial equations, MATH2001 lecture note (2012), The Univ. of Hong Kong. (http://hkmath.hku.hk/course/MATH2001)
4 Peter Pesic, Abel's proof: An essay on the sources and meaning of mathematical unsolvability, The MIT press, 2003.
5 Kragh H. Sorensen, Niels Henrik Abel and the theory of equations, 1999. (http://www.henrikkragh.dk/pdf/part199911g.pdf)
6 John Stillwell, Mathematics and its history, Undergraduate Texts in Mathematics, 3rd ed., Springer, 2010.
7 Jeff Suzuki, Lagrange's proof of the fundamental theorem of algebra, MAA Monthly 113(8) (Oct. 2006), 705-714.   DOI
8 Jean-Pierre Tignol, Galois' theory of algebraic equations, World Scientific, 2001.
9 Ehrenfried W. von Tschirnhaus, A method for removing all intermediate terms from a given equation, Acta Eruditorum (May 1683), 204-207. Translated by R. F. Green, ACM SIGSAM Bulletin 37(1) (2003).
10 Heine J.Barnett, Abstract awakening in algebra: Early group theory in the work of Lagrange, Cauchy, and Cayley, Lecture note, Colorado State University-Pueblo, 2011.
11 William Dunham, Journey through genius: The great theorems of mathematics, The Wiley Science Editions, John Wiley and Sons, Inc., 1990.
12 Greg St. George, Symmetric polynomials in the work of Newton and Lagrange, Math. Mag. 76(5) (2003), 372-378.