Niels Abel and Evariste Galois: Life and Theories

Niels Abel's life was full of poverty. After the death of his father, who was a Protestant minister, in 1820 Abel had the responsibility of supporting his mother and family. Abel's teacher, Holmboe, raised money from his colleagues to enable Abel to attend Christiania University. He entered the university in 1821 and graduated in 1822. In 1823, Abel published papers on functional equations and integrals. Abel gives the first solution of an integral equation in it. In 1824 he proved the impossibility of solving algebraically the general equation of the fifth degree.

His major work, Recherches sur les fonctions elliptiques, was published in 1827.After visiting Paris, he returned to Norway in debt. While in Paris, he visited a doctor who told him he had tuberculosis. He continued writing papers on equation theory and elliptic functions important in the development of the whole theory. Abel travelled to visit his fiancee for Christmas 1828 in Froland. He became ill on the journey and died a couple of months later. He later became famous for his contributions to group theory.  Galois's life was controlled by politics and mathematics. In 1829 he published his first paper on continued fractions, followed by a paper that dealt with the impossibility of solving the general quintic equation by radicals. This led to his theory, a branch of mathematics dealing with the general solution of equations. Famous for his contributions to group theory, he produced a method of figuring out when a general equation could be solved by radicals. This theory solved many unanswered questions including the impossibility of trisecting the angle and squaring the circle. He introduced the term 'group' when he considered the group of permutations of the roots of an equation.

Galois's work made an important contribution to the transition from classical to modern algebra. Having spent some time in prison for political offenses, he was killed in a duel at the age of 21 shortly after his release from prison.