*(English)*Zbl 1065.11001

This article describes Artin’s research leading up to what nowadays has become the center of class field theory: Artin’s reciprocity law.

Artin’s dissertation [JFM 48.0189.02] on quadratic extensions of ${\mathbb{F}}_{p}\left[X\right]$ builds on previous work by Gauss, Dedekind and Kornblum [Math. Z. 5, 100–111 (1919; JFM 47.0154.02)]. Artin’s next topic was the question whether, for extensions $L/K$ of number fields, the quotient ${\zeta}_{L}\left(s\right)/{\zeta}_{K}\left(s\right)$ of zeta functions was an entire function [Math. Ann. 89, 147–156 (1923; JFM 49.0123.02)]. Artin’s next article “On a new kind of $L$-functions” [Hamb. Math. Abh. 3, 89–108 (1923; JFM 49.0123.01)] already contained his reciprocity law as a conjecture – he obtained the proof only in 1927 [Abh. Hamb. 5, 353–363 (1927; JFM 53.0144.04)], using an idea due to Chebotarev.