Cauchy's research on substitutions

N. V. Ingtem

Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 198-210.

Résumé

The article is devoted to the introduction and formation of the term and the action symbol "substitution". In mathematical research before Lagrange, it was never practiced to rearrange independent variables contained in a given function. For the first time this technique is found in Lagrange's work of 1771, devoted to the algebraic solution of equations. Vandermond, who had published his work in the same year 1771, has expressed the idea of the need to introduce notations that simplify calculations and the perception of operations on root functions. However, the introduced designations were not easy to understand and became more complicated with increasing the degree of the equation. Ruffini's works, published from 1799 to 1813, aimed to prove the impossibility of solving the equation of the $5$th degree and are, in fact, a study of the symmetric group represented by the values of the root function in the form of all possible permutations of these roots. During these researches, he proves that the group $S_5$ does not contain subgroups of the index $3$, $4$ or $8$. However, just like Lagrange, Ruffini uses complex cumbersome expressions. Cauchy, dealing with issues of combinatorial analysis, tried to generalize the result obtained by Ruffini to equations of arbitrary degree. Working on the determination of the limits that a function of $n$ variables can take, Cauchy has invented a new research tool, which later became an independent theory. This was the the substitution group theory.

Détail

Keywords: permutation, substitution, combination types, partial types, symmetric functions, function index, product of substitutions.

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