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Galois, Évariste

theory equations roots École

[galwah] (1811–32) French mathematician: founded modern group theory.

Galois was born with a revolutionary spirit, politically and mathematically, and at an early age discovered he had a talent for original work in mathematics. He was taught by his mother until the age of 12, and at school found interest only in exploring books by creative mathematicians such as A-M Legendre (1752–1833), . Twice (in 1827 and 1829) Galois took the entrance examinations for the École Polytechnique (which was already one of the foremost colleges for science and mathematics) but failed on each occasion. At 17 he submitted a paper to the French Académie des Sciences via , but this was lost. Following his father’s suicide Galois entered the École Normale Supérieure to train as a teacher. During 1830 he wrote three papers breaking new ground in the theory of algebraic equations and submitted them to the Academy; they too were lost.

In the political turmoil following the 1830 revolution and Charles X’s abdication Galois chided the staff and students of the École for their lack of backbone and he was expelled. A paper on the general solution of equations (now called Galois theory) was sent via to the Academy, but was described as ‘incomprehensible’. In 1831 he was arrested twice, for a speech against the king and for wearing an illegal uniform and carrying arms, and received 6 months’ imprisonment. Released on parole, Galois was soon challenged to a duel by political opponents. He spent the night feverishly sketching out in a letter as many of his mathematical discoveries as he could, occasionally breaking off to scribble in the margin ‘I have not time’. At dawn he received a pistol shot through the stomach, and having been left where he fell was found by a passing peasant. Following his death from peritonitis 8 days later he was buried in the common ditch of South Cemetery, aged 20.

The letter and some unpublished papers were discovered by 14 years later, and are regarded as having founded (together with Abel’s work) modern group theory. It outlines his work on elliptic integrals and sets out a theory of the solutions (roots) of equations by considering the properties of permutations of the roots. If the roots obey the same relations after permutation they form what is now called a Galois group, and this gives information on the solvability of the equations.

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