"Galois' Theory of Algebraic Equations" gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the 19th century. The main emphasis is placed on equations of at least the third degree, ie. on the developments during the period from the 16th to the 19th century. The appropriate parts of wo...
"Galois' Theory of Algebraic Equations" gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the 19th century. The main emphasis is placed on equations of at least the third degree, ie. on the developments during the period from the 16th to the 19th century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as "group" and "field". A brief discussion on the fundamental theorems of modern Galois theory is included. Complete proofs of the quoted results are provided, but the material has been organized in such a way that the most technical details can be skipped by readers how are interested primarily in a broad survey of the theory. This book should appeal to both undergraduate and graduate students in mathematics and the history of science, and also to teachers and mathematicians who wish to obtain an historical perspective of the field. The text has been designed to be self-contained, but some familiarity with basic mathematical structures and with some elementary notions of linear algebra is desirable for a good understanding of the technical discussions in the later chapters.
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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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还没人写过短评呢
还没人写过短评呢