Eisenstein's footnote
The ``solution'' of the quintic equation represents one of the high points of 19th century mathematics, and it is indelibly attached to the names of Hermite and Klein. It is less well known that G. Eisenstein had given a ``solution'' 14 years before Hermite. He gave it only as a footnote in one of his early papers and hence the title of the paper. His solution consists of two steps. First of all Eisenstein uses the Bring-Jerrard reduction (to an equation of the form \(X^5+ X-\lambda=0\)). The author argues, convincingly, that Eisenstein had become aware of Jerrard's work during a visit to Dublin when he met W. R. Hamilton. (Bring was a Swedish mathematician; the reduction of the quintic was given in his thesis of 1786, but it was apparently completely forgotten.) Eisenstein then gives an elegant power series expansion of a root of the equation for \(\lambda\) small and \(X\) close to \(\lambda\). This one would now derive by means of the Lagrange-Bürmann theorem. Eisenstein had his own method, and the author remarks that Lambert had discovered essentially the same results in 1758 -- but then the Bring-Jerrard reduction was unknown. Eisenstein does not make any connection with the theory of elliptic functions; indeed this would have been more or less impossible at the time. In this paper the author gives an account both of the history and of the mathematics of Eisenstein's discovery.
scientific article; zbMATH DE number 851678