He was the first to use the word "group" as a technical term in mathematics to represent a group of permutations. His work on equation theory was submitted to the Academy and was reviewed by Poisson, who did not understand it. It was resubmitted again in shorter form. The truth and importance of the work was not confirmed during his lifetime. His work laid the foundation for [Galois theory]?, a major branch of abstract algebra.
He was a staunch Republican, famous for having toasted Louis Philippe with a dagger above his cup, which leads some to believe that his death in a duel was set up by the secret police.
The night before a duel he supposedly fought in order to defend the honor of a woman, he 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.
In his final papers he outlined the rough edges of some work he had been doing in analysis and annotated a copy of the manuscript submitted to the academy.
The next day he was shot in the abdomen and died the following day in hospital (probably of peritonitis) after refusing the offices of a priest.
His work was not understood until 1843 when Louiville reviewed his manuscript and declared that he had indeed solved the problem first proposed by Abel.
The manuscript was finally published in the October-November 1846 issue of the Journal des mathématiques pures et appliquées.
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