라그랑주의 방정식론
Lagrange and Polynomial Equations
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초록
16세기에 3차, 4차 방정식의 대수적인 해의 공식이 구해진 후, 5차 이상의 방정식에 대하여서도 해의공식을 구하는 것이 중요 문제가 되었고 이를 위한 노력이 200여년간 계속되었다. 라그랑주는 이전에알려져있던 방정식의 해법을 면밀히 조사하여 일반적인 방정식에 적용할 수 있는 하나의 원리를찾고자 하였다. 이 과정에서 방정식의 해들의 순열과 보조 방정식과의 관계를 알아내었는데, 해들의순열은 나중에 아벨의 증명과 갈루아 이론에 핵심적인 아이디어가 되었다. 이 논문에서는 라그랑주를비롯한 여러 수학자들이 제시한 방정식의 해법을 자세히 분석하여 방정식론에 있어서 라그랑주가미친 영향과 그 의의를 살펴본다.
Abstract
After algebraic expressions for the roots of 3rd and 4th degree polynomial equationswere given in the mid 16th century, seeking such a formula for the 5th andgreater degree equations had been one main problem for algebraists for almost 200years. Lagrange made careful and thorough investigation of various solving methodsfor equations with the purpose of finding a principle which could be applicableto general equations. In the process of doing this, he found a relation betweenthe roots of the original equation and its auxiliary equation using permutations ofthe roots. Lagrange’s ingenious idea of using permutations of roots of the originalequation is regarded as the key factor of the Abel’s proof of unsolvability byradicals of general 5th degree equations and of Galois’ theory as well. This paperintends to examine Lagrange’s contribution in the theory of polynomial equations,providing a detailed analysis of various solving methods of Lagrange and othersbefore him.
참고문헌 (12)
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