Cauchy, a forgotten pioneer

Because Augustin-Louis Cauchy did not take a direct interest in solving algebraic equations, he is an overlooked figure in the history of group theory. Yet his research on permutations provided valuable tools for those who worked on Galois theory.

Written by

FRANCOIS LAVALLOU

PublishedMay 18, 2022UpdatedMay 16, 20261 min read

In the 1770s, mathematicians Edward Waring (1736–1798), Joseph-Louis Lagrange (1736–1813) and Alexandre-Théophile Vandermonde (1735–1796) developed the theory of permutations, viewed as rearrangements of the roots of an equation. This became a central tool in their study of the algebraic theory of equations. Waring failed to achieve lasting recognition despite his numerous but obscure works, while Vandermonde took his first steps in mathematics with symmetric functions; it was Lagrange who laid the foundations of Galois theory (see our special issue 80, Les Groupes (Groups), 2021). Permutation theory then advanced in the early 19^th century through the work of Paolo Ruffini (1765–1822) and Augustin-Louis Cauchy (1789–1857).

####

Permutations of three apples (one yellow, one green and one red).

Substitutions as Cauchy saw them ---------------------------------

Discussion

Comments

Join the discussion

No comments yet. Be the first to respond.

On the same theme

Articles recommended for you.

Math for everyone

Counting by substitution

At barely 25, Cauchy puts the finishing touches to a paper on symmetric functions that is eventually published as two articles. In it, he introduces new concepts, notation and methods, and creates the calculus of substitutions, which will play a central role in the development of group theory.

FRANCOIS LAVALLOUAug 24, 2024

Math for everyone

Origins of group theory: Cauchy and Galois | Tangente

Cauchy the analyst is well known; Cauchy the algebraist, much less so. Cauchy's contribution to group theory long went unrecognized, even though his research on algebraic structures was highly influential.

FRANCOIS LAVALLOUAug 25, 2024

History and Culture

Beyond Lagrange's memoir

As a teenager, Galois read Legendre and Lagrange, followed by Gauss and Cauchy. He often cites the latter two, but rarely Lagrange. Galois was clearly influenced by Lagrange's ideas; he would, however, go much further, benefiting from all the advances made since 1771.

MARC THIERRYMay 19, 2022

History and Culture

Early formalizations

Évariste Galois's tragic death lent an epic quality to the introduction of the group concept in mathematics. What followed is less familiar but fascinating, culminating in brilliant theorems that are still taught today. Sixty years later, the notion of a group was finally established.

BERTRAND HAUCHECORNENov 17, 2021