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Budding mathematician

In 1828, Galois 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. In that same year, he entered the École Normale (then known as l'École préparatoire), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.

In the following year Galois's first paper, on continued fractions, was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences. Augustin-Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear. However, in spite of many claims to the contrary, it is widely held that Cauchy recognized the importance of Galois's work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the academy's Grand Prize in Mathematics. Cauchy, an eminent mathematician of the time though with political views that were diametrically opposed to those of Galois, considered Galois's work to be a likely winner.

On 28 July 1829, Galois's father died by suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt to enter the Polytechnique and failed yet again. It is undisputed that Galois was more than qualified; accounts differ on why he failed. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois. The recent death of his father may have also influenced his behavior.

Having been denied admission to the École polytechnique, Galois took the Baccalaureate examinations in order to enter the École normale. He passed, receiving his degree on 29 December 1829. 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."

Galois submitted his memoir on equation theory several times, but it was never published in his lifetime. 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, to be considered for the Grand Prix of the academy. Unfortunately, Fourier died soon after, and the memoir was lost. The prize would be awarded that year to Niels Henrik Abel posthumously and also to Carl Gustav Jacob Jacobi. Despite the lost memoir, Galois published three papers that year. One laid the foundations for Galois theory. The second was about the numerical resolution of equations (root finding in modern terminology). The third was an important one in number theory, in which the concept of a finite field was first articulated.