Daisuke Ishii
Jan 7, 2018 · 4 min read

From Wikipedia;

Évariste Galois (French: [evaʁist ɡaˈlwa]; 25 October 1811–31 May 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem standing for 350 years. His work laid the foundations for Galois theory and group theory,[1] two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel.

Wikipediaより;

数学者として10代のうちにガロア理論の構成要素である体論群論の先見的な研究を行った。ガロアはガロア理論を用い、ニールス・アーベルによる「五次以上の方程式には一般的な代数的解の公式がない」という定理(アーベル-ルフィニの定理)の証明を大幅に簡略化し、また、より一般にどんな場合に与えられた方程式が代数的な解の表示を持つかについての特徴付けを与えた。また、数学史上初めてカテゴリー論的操作によって自らの理論の基礎を構築している。

群論は数学でも重要だが、数学以外、例えば物理では相対性理論量子力学などを厳密に(形式的に)記述するツールとして用いられる。また、計算機科学、特に理論計算機科学においてガロア体、特に位数2のガロア体 F2 は最も多用される数学的ツールのひとつである。

このように代数学で重要な役割を果たすガロア理論は、現代数学の扉を開くとともに、20世紀、21世紀科学のあらゆる分野に絶大な影響を与えている。しかし、ガロアの業績の真実と重要性、先見性は当時世界最高の研究機関であったパリ科学アカデミーを初め、カール・ガウスオーギュスタン・コーシーカール・ヤコビと言った歴史に名を残した同時代の大数学者達にさえ理解されず、生前に評価されることはなかった[1]。群論の基礎概念とも言える集合論ゲオルク・カントールによって提唱され、ガロア理論へと通じる数学領域が構築されるのでさえ、ガロアによるガロア理論構築の50年も後のことである。

ガロアの遺書となった友人宛の手紙には、後の数学者たちにとって永年の研究対象となる理論に対する着想が「僕にはもう時間がない」 (je n’ai pas le temps) という言葉と共に書き綴られている。例えば代数的には解けない5次以上の方程式の解を与える、楕円モジュラー関数による超越的解の公式の存在を予言し、そのアイデアを記している。なお、この手法はガロアの死後50年の時を経てシャルル・エルミートによって確立される。

From YouTube;

From Wikipedia;

From the closing lines of a letter from Galois to his friend Auguste Chevalier, dated May 29, 1832, two days before Galois’ death:[21]

Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l’importance des théorèmes.

Après cela, il y aura, j’espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

(Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)

Within the 60 or so pages of Galois’ collected works are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.[26][27] His work has been compared to that of Niels Henrik Abel, another mathematician who died at a very young age, and much of their work had significant overlap.

Algebra

While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group (in French groupe) in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as group theory. He developed the concept that is today known as a normal subgroup. He called the decomposition of a group into its left and right cosets a proper decomposition if the left and right cosets coincide, which is what today is known as a normal subgroup.[21]He also introduced the concept of a finite field (also known as a Galois field in his honor), in essentially the same form as it is understood today.[10]

In his last letter to Chevalier[21] and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields:

Galois theory

Main article: Galois theory

Galois’ most significant contribution to mathematics is his development of Galois theory. He realized that the algebraic solution to a polynomialequation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, or its Galois group is solvable. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations to which Galois originally applied it.[26]

Analysis

Galois also made some contributions to the theory of Abelian integrals and continued fractions.

As written in his last letter,[21] Galois passed from the study of elliptic functions to consideration of the integrals of the most general algebraic differentials, today called Abelian integrals. He classified these integrals into three categories.

日本語の解説書;

ガロアの研究;

Team AI Research

Research on Latest Machine Learning Technology. http://www.team-ai.com/ 機械学習の最新技術情報を分析しています

Daisuke Ishii

Written by

Founder of Jenio Inc & Machine Learning Community “Team AI” in Tokyo. http://www.team-ai.com/

Team AI Research

Research on Latest Machine Learning Technology. http://www.team-ai.com/ 機械学習の最新技術情報を分析しています

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