Adolf Hurwitz and the Fundamental Theorem of Galois Theorie: The K\"onigsberg Lectures of 1890-1891

TL;DR

This paper analyzes Adolf Hurwitz's 1890-1891 lecture course on Galois theory, highlighting his clear presentation and detailed proof of the fundamental theorem using substitution language.

Contribution

It provides a detailed reconstruction of Hurwitz's original argument and contextualizes his work within the historical development of Galois theory.

Findings

Hurwitz's lectures offer a clear presentation of Galois ideas.

Hurwitz's proof uses substitution language effectively.

Historical context enriches understanding of Galois theory's development.

Abstract

In the winter semester of 1890--1891 Adolf Hurwitz delivered a lecture course at the Albertina University in K\"onigsberg entitled -Theorie der algebraischen Gleichungen-. These lectures contain a particularly clear presentation of the ideas of Evariste Galois and, in particular, a proof of the fundamental theorem of Galois theory formulated in the language of substitutions. The present paper analyzes Hurwitz's treatment of this result on the basis of his lecture notes preserved in the ETH Library in Zurich (Hs 582:66), together with material from his Mathematisches Tagebuch 23 (Hs 582:23). After placing the K\"onigsberg lectures in their historical context, we give an overview of their mathematical content and reconstruct in detail Hurwitz's argument leading to the fundamental theorem.