Galois

The development of abstract algebra brought with itself group theory, rings and fields, Galois theory.

Galois showed just before his untimely death that these efforts were largely wasted.

This is a result of Galois theory ( see Quintic equations and the Abel – Ruffini theorem ).

Field automorphisms are important to the theory of field extensions, in particular Galois extensions.

In the case of a Galois extension L / K the subgroup of all automorphisms of L fixing K pointwise is called the Galois group of the extension.

* 1832 – Évariste Galois is released from prison.

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

He also provided an algebraic definition of fundamental groups of schemes and more generally the main structures of a categorical Galois theory.

Esquisse d ’ un Programme was published in the two-volume proceedings Geometric Galois Actions ( Cambridge University Press, 1997 ).

La Longue Marche à travers la théorie de Galois Long March Through Galois Theory is an approximately 1600-page handwritten manuscript produced by Grothendieck during the years 1980 – 1981, containing many of the ideas leading to the Esquisse d ' un programme ( see below, and also a more detailed entry ), and in particular studying the Teichmüller theory.

In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

# Motives and the motivic Galois group ( and Grothendieck categories )

# Yoga of anabelian geometry and Galois – Teichmüller theory.

* Grothendieck's Galois theory

In general, the absolute Galois group of K is the Galois group of K < sup > sep </ sup > over K.

* A profinite group ( e. g., Galois group ) is compact.

* Artin conductor, an ideal or number associated to a representation of a Galois group of a local or global field

Typical of the courses he teaches is his seminar " Group Theory and Galois Theory Visualized ", in which abstract mathematical ideas are rendered as concretely as possible.

* Galois extension

* Splitting of prime ideals in Galois extensions

Évariste Galois () ( 25 October 1811 – 31 May 1832 ) was a French mathematician born in Bourg-la-Reine.

His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections.