Galois group

…About half a century later, N.H. Abel and E. Galois developed the idea further, with Abel first investigating polynomials that could be solved algebraically (those that start from the coefficients and have roots that can be obtained by arithmetic operations and taking the roots of powers), and showing that there are general quintic polynomials that cannot be solved algebraically. Galois went further, considering groups of polynomials (what we would call Galois groups today), and also defining rooted fields and their intermediate fields, elucidating the relationship between the structure of Galois groups and intermediate fields. It was also Galois who defined normal subgroups, which will be discussed later, and Galois' research can be said to be the beginning of group theory. …

*Some of the terminology explanations that mention "Galois group" are listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

…約半世紀後にN.H.アーベルとE.ガロアがその考えを進展させて，アーベルが，まず代数的に解ける多項式(係数から出発して，根が四則算法とべき根をとる演算とで得られるもの)を調べ，一般の五次多項式には代数的には解けないものがあることを示した。ガロアはさらに進んで，多項式の群(今日の言葉でいうガロア群)を考え，また，根をつけた体やその中間体も定義して，ガロア群の構造と中間体との関係を解明した。また，後に述べる正規部分群を定義したのもガロアであり，このガロアの研究が群論の始まりといえる。…

※「ガロア群」について言及している用語解説の一部を掲載しています。

出典｜株式会社平凡社世界大百科事典 第２版について | 情報