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.

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

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

Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie ( born Demante ).

At the age of 10, Galois was offered a place at the college of Reims, but his mother preferred to keep him at home.

It is undisputed that Galois was more than qualified ; however, accounts differ on why he failed.

Although the Gazettes editor omitted the signature for publication, Galois was expelled.

Due to controversy surrounding the unit, soon after Galois became a member, on 31 December 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government.

The proceedings grew riotous, and Galois proposed a toast to King Louis-Philippe with a dagger above his cup, which was interpreted as a threat against the king's life.

On the following Bastille Day, Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger.

Évariste Galois was buried in a common grave and the exact location is still unknown.

" While Poisson's report was made before Galois ' Bastille Day arrest, it took until October to reach Galois in prison.

Some archival investigation on the original letters suggests that the woman of romantic interest was a Mademoiselle Stéphanie-Félicie Poterin du Motel, the daughter of the physician at the hostel where Galois stayed during the last months of his life.

Much more detailed speculation based on these scant historical details has been interpolated by many of Galois ' biographers ( most notably by Eric Temple Bell in Men of Mathematics ), such as the frequently repeated speculation that the entire incident was stage-managed by the police and royalist factions to eliminate a political enemy.

As to his opponent in the duel, Alexandre Dumas names Pescheux d ' Herbinville, one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois ' first arrest and du Motel's fiancé.

However, Dumas is alone in this assertion, and extant newspaper clippings from only a few days after the duel give a description of his opponent that more accurately applies to one of Galois ' Republican friends, most probably Ernest Duchatelet, who was imprisoned with Galois on the same charges.

Whatever the reasons behind the duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament, the famous letter to Auguste Chevalier outlining his ideas, and three attached manuscripts.

On 2 June, Évariste Galois was buried in a common grave of the Montparnasse cemetery whose exact location is unknown.

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.

This conjecture is also supported by other letters Galois later wrote to his friends the night before he died.

In 1853 he wrote a memoir on the algebraic solvability of equations extending the work of Évariste Galois on the theory of equations.

Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory.

It is unsurprising, in the light of his character and situation at the time, that Galois reacted violently to the rejection letter, and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier.

The general linear group over a prime field, GL ( ν, p ), was constructed and its order computed by Évariste Galois in 1832, in his last letter ( to Chevalier ) and second ( of three ) attached manuscripts, which he used in the context of studying the Galois group of the general equation of order p < sup > ν </ sup >.

Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3 ; this is contained in his last letter to Chevalier.

In the same letter and attached manuscripts, Galois also constructed the general linear group over a prime field, GL ( ν, p ), in studying the Galois group of the general equation of degree p < sup > ν </ sup >.

By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Galois.

This terminology has been introduced in his last letter by Évariste Galois who called ( in French ) equation primitive an equation whose Galois group is primitive.

While their counterparts at Polytechnique were making history in the streets during les Trois Glorieuses, Galois and all the other students at the École Normale were locked in by the school's director.

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.

More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois.

Despite the lost memoir, Galois published three papers that year, one of which laid the foundations for Galois theory.

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.

The study of field extensions ( and polynomials which give rise to them ) via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

To do this, he invented ( independently of Galois ) an extremely important branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics as well.

Specifically, if L / K is a Galois extension, we consider the group G = Gal ( L / K ) consisting of all field automorphisms of L which keep all elements of K fixed.

Waterhouse showed that every profinite group is isomorphic to one arising from the Galois theory of some field K ; but one cannot ( yet ) control which field K will be in this case.

In fact, for many fields K one does not know in general precisely which finite groups occur as Galois groups over K. This is the inverse Galois problem for a field K. ( For some fields K the inverse Galois problem is settled, such as the field of rational functions in one variable over the complex numbers.

There are two threads in the history of finite simple groups – the discovery and construction of specific simple groups and families, which took place from the work of Galois in the 1820s to the construction of the Monster in 1981 ; and proof that this list was complete, which began in the 19th century, most significantly took place 1955 through 1983 ( when victory was initially declared ), but was only generally agreed to be finished in 2004., work on improving the proofs and understanding continues ; see for 19th century history of simple groups.

Simple groups have been studied at least since early Galois theory, where Évariste Galois realized that the fact that the alternating groups on five or more points was simple ( and hence not solvable ), which he proved in 1831, was the reason that one could not solve the quintic in radicals.

In some cases, it is possible to define abstractions using Galois connections ( α, γ ) where α is from L to L ′ and γ is from L ′ to L. This supposes the existence of best abstractions, which is not necessarily the case.

Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.