Yet, he had certainly noticed that the relations were not preserved by every permutation of the roots. The existence of a cyclic permutation that does preserve the relations is a very remarkable, and quite mysterious, property of cyclotomic equations, which should have awaken Vandermonde’s curiosity. If he had investigated this property, he could have developed the theory of cyclotomy about thirty years before Gauss.
Moreover, Vandermonde had pinpointed the very basic idea of Galois theory: in order to determine the “structure” of an equation, deciding eventually whether it is solvable by radicals, and more generally to evaluate its difficulty, one has to look at the permutations of the roots; but one needs only to consider those permutations that preserve the relations between the roots.... (查看原文)
Since, to solve the equation
X^m − X^(m−1) − (m− 1)X^(m−2) + etc. =0,
the question is at most to determine (article VI) the quantity which is indifferently one of its roots, and by no means to arrange it in such a way that it be indifferent to interchange the roots among themselves, this solution will always be very easy [81, p. 416]. (查看原文)
In spite of the title, the main subject of these lectures is not algebra, even less history, as one could conclude from a glance over the table of contents, but methodology. Their aim is to convey to the audience, which originally consisted of undergraduate...
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Nirvana
World Scientific Pub Co Inc2001版
还没人写过短评呢
还没人写过短评呢