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On the following Bastille Day (14 July 1831), 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. Galois returned to mathematics after his expulsion from the École Normale, although he continued to spend time in political activities.After his expulsion became official in January 1831, he attempted to start a private class in advanced algebra which attracted some interest, but this waned, as it seemed that his political activism had priority.His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research." He submitted his memoir on equation theory several times, but it was never published in his lifetime due to various events.Though his first attempt was refused by Cauchy, in February 1830 following Cauchy's suggestion he submitted it to the Academy's secretary Joseph Fourier, The prize would be awarded that year to Niels Henrik Abel posthumously and also to Carl Gustav Jacob Jacobi.At 15, he was reading the original papers of Joseph-Louis Lagrange, such as the Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory, and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.In 1828, he attempted the entrance examination for the École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination.More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois.

It is undisputed that Galois was more than qualified; however, accounts differ on why he failed.Although the Gazette's editor omitted the signature for publication, Galois was expelled.Although his expulsion would have formally taken effect on 4 January 1831, Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard.He soon became bored with his studies and, at the age of 14, he began to take a serious interest in mathematics.He found a copy of Adrien-Marie Legendre's Éléments de Géométrie, which, it is said, he read "like a novel" and mastered at the first reading.

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