Galois

Galois returned to mathematics after his expulsion from the École Normale, although he continued to spend time in political activities.

Around 4 July, Poisson declared Galois ' work " incomprehensible ", declaring that " argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor "; however, the rejection report ends on an encouraging note: " We would then suggest that the author should publish the whole of his work in order to form a definitive opinion.

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

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.

Apparently, however, Galois did not ignore Poisson's advice, began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832.

Galois ' fatal duel took place on 30 May.

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.

Fragments of letters from her copied by Galois himself ( with many portions either obliterated, such as her name, or deliberately omitted ) are available.

du Motel had confided some of her troubles to Galois, and this might have prompted him to provoke the duel himself on her behalf.

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

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.

" However, the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated.

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

Galois ' mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript and declared it sound.

Although Abel had already proved the impossibility of a " quintic formula " by radicals in 1824 and Ruffini had published a solution in 1799 that turned out to be flawed, Galois ' methods led to deeper research in what is now called Galois theory.

Unsurprisingly, Galois ' collected works amount to only some 60 pages, but within them are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.

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 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.