Research On The Cyclotomic Theory In Gauss's Disquisitiones Arithmeticae

Gauss's Disquisitiones Arithmeticae is a classical work in the history of mathematics. Its birth not only marks the beginning of modern number theory, but also is an important turning point in the history of algebra. We should reexamine the Gauss's work under the circumstance of the historical development of Galois Theory. In this way we will find that the theory of Gauss's cyclotomic equation is an indispensable part to Galois Theory. In fact, cyclotomic equation is related to the Construction of Regular Polygons. Moreover, it is also a kind of special equation which could be solved by radicals. Thus, it has a high place in the history of "Galois Theory". This thesis mainly focuses on the following aspects:(1) In the first part of this thesis, this writer systematically reviews the historical development of "Galois Theory". Some important mathematician's works are discussed.(2) Because of the importance of Lagrange's contribution, a detailed introduction to the strategy of Lagrange on the problem of solvability by radicals is presented.(3) After an introduction to the great work of Gauss and a detailed study of chapterâ…¦of Disquisitiones Arithmeticae, this writer finds out the way Gauss solves the cyclotomy equation, as well as the relation between Gauss's idea and Lagrange's strategy. In addition to that, Gauss's influence on the descendants, such as Abel, Galois is also explored.