Acceptance of Galois Theory

Galois’ work was the end of one main line of algebra—solving equations by radical methods—and the beginning of a new line—the study of abstract structures. The work on the sequence started by Lagrange and Ruffini received further inspiration in 1815 from the famous French mathematician Augustin-Louis Cauchy. In work after 1844, Cauchy systematized much of this knowledge and introduced basic concepts. For example, setting

Cycle of change

Cauchy indicated in cycle notation as (ab)(ced), meaning that the sequence was achieved by separate cycles from a to b (and back to a) and c to e to d (and back to c).

A series of unusual and unfortunate events involving the most important contemporary French mathematicians prevented Galois’ theorems from being published for a long time. It was not until 1846 that Joseph Liouville first edited and published his authoritative Journal de Mathematics Pures et Appliques, the important treatise in which Galois presented his central ideas and which was rejected by the Paris Academy in 1831. In Germany, Leopold Kronecker applied some of these ideas to number theory in 1853, and Richard Dedekind lectured on Galois theory in 1856. However, the influence of the theory was still low at the time.

A turning point came with the publication of Traité des substitutions et des equations algebriques (1870; “Treatise on Substitutions and Algebraic Equations”) by the French mathematician Camille Jordan. In his book and papers, Jordan elaborated an abstract theory of groups of order, in which algebraic equations serve only as an illustrative application of the theory. In particular, Jordan’s treatise was the first book on group theory and served as the basis for the concept of Galois theory as a study of the interrelationships between corresponding Galois groups of extensions of fields and equations. New abstract approach to algebra in the 1920s. Important contributions to the development of this approach to Galois theory came variously from Enrico Betti (1823–92) in Italy and Dedekind, Heinrich Weber (1842–1913) and Emil Arten (1898–1962) in Germany.