ダウンロード ラグランジュによる2～4次方程式の解法とガロア群: 良く分るガロア群―真説ガロア群 pdf 本

本著では，ガロア理論を，ガロアが辿ったと考えられる時系列順に，ガロアの，死亡後の1846 年に発刊された論文「代数方程式の累乗根で解ける条件に付いて」の内容に先輩著者等による証明等も参考・引用しながら，著者自身の研究の結果も付けて記述した。ガロアが「有理式」と言っている用語は，今日我々が通常言っている「有理式」よりは意味が広く（汎く），現在で言う「方程式の全ての根の，１つの有理式」の外に，この有理式の，方程式の根の置換による作用後の式も「有理式」（正確には「有理的式」）と言っている。ラグランジュによる，n 次代数方程式の根の表式は，n 個の代数方程式の根を， (n 1) 個の，その原代数方程式の全ての根の１次式の線形和に換えている（これを本著では未知数変換ではなく，未知数転換と言っている）。この未知数転換後の未知数の方程式に付いて解ければ，原方程式と未知数転換後の方程式との結合関係式が定まっている場合には，原方程式が解けることになる。その手順を本著では詳細に述べた。また，代数方程式の根が形成されていく過程を，ガロア群の元の個数の減少と，全ての根から成る１つの有理式 (従って根を構成する全ての有理式) が確定する体の拡張との対応を示しながら詳細に示した。最後に本著のまとめを述べた。Overview of this book (Solutions of Algebraic Equations and Galois Group) It is interesting to know how Galois group concerns solutions of algebraic equations. Galois, mathematician of France, who died tenderly by a duel in 1832, contrived and created Galois group consisting of the substitution of roots of an algebraic equation and developed the theory of "conditions under which algebraic equations can be solved using only rational functions and their roots", based on theories about solutions of algebraic equations that Lagrange, Euler, Gauss, Abel and others built and developedFollowing the time series where Galois is thought to have pursed, this book describes Galois theory in the paper published in 1846 after Galois's death, using referring and quoting proofs of contents of papers by senior authors and others, and adding expressing results in a research by the author, himself.The term of a "rational formula" which Galois said has wider meaning than a "rational formula" which we say today generally.Galois says that it is, also, a formula after an operation by the substitution of roots of an equation (It is a "formula like the rationality" correctly.), in addition to a "rational formula of all roots of an equation" that we say at present.By Lagrange Expression of the whole root of an algebraic equation, the whole root of an algebraic equation of -th degree is converted to that of an algebraic equation of -th degree, consisting of a linear summation of the 1st degree formulae of all roots of the algebraic equation (In this book, it is called to be not the unknown quantity transform but unknown quantity conversion.).If the equation with unknown quantities after this unknown quantityconversion can be solved, then the first equation will be able to be solved, in the case where the combination formula between the first equation and that after an unknown quantity conversion will be decided.This book describes the procedure in detail. The author also shows, in detail, the process in which the roots of an algebraic equation are formed, while showing the correspondence between the decrease of an original number of Galois group and the extension of the body whose rational forms of the roots are fixed.Finally, this book describes the summary of this work.