17 Facts About Galois theory

In mathematics, Galois theory, originally introduced by Evariste Galois, provides a connection between field theory and group theory.

Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated, and characterizing the regular polygons that are constructible.

The Galois theory took longer to become popular among mathematicians and to be well understood.

Galois theory has been generalized to Galois connections and Grothendieck's Galois theory.

Birth and development of Galois theory was caused by the following question, which was one of the main open mathematical questions until the beginning of 19th century:.

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Galois' theory provides a much more complete answer to this question, by explaining why it is possible to solve some equations, including all those of degree four or lower, in the above manner, and why it is not possible for most equations of degree five or higher.

Galois' theory gives a clear insight into questions concerning problems in compass and straightedge construction.

Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are the elementary symmetric polynomials in the roots.

Galois theory was the first who discovered the rules for summing the powers of the roots of any equation.

Galois theory's solution contained a gap, which Cauchy considered minor, though this was not patched until the work of the Norwegian mathematician Niels Henrik Abel, who published a proof in 1824, thus establishing the Abel–Ruffini theorem.

In 1830 Galois submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals; Galois' paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients.

Galois theory then died in a duel in 1832, and his paper, "Memoire sur les conditions de resolubilite des equations par radicaux", remained unpublished until 1846 when it was published by Joseph Liouville accompanied by some of his own explanations.

Galois' theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it.

The central idea of Galois' theory is to consider permutations of the roots such that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted.

Originally, the Galois theory had been developed for algebraic equations whose coefficients are rational numbers.

One of the great triumphs of Galois Theory was the proof that for every, there exist polynomials of degree which are not solvable by radicals, and a systematic way for testing whether a specific polynomial is solvable by radicals.

Inverse Galois theory problem is to find a field extension with a given Galois theory group.