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On the history of the Artin reciprocity law in Abelian extensions of algebraic number fields: how Artin was led to his reciprocity law. (English) Zbl 1065.11001
Laudal, Olav Arnfinn (ed.) et al., The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer (ISBN 3-540-43826-2/hbk). 267-294 (2004).

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 𝔽 p [X] 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 ζ L (s)/ζ K (s) 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.

MSC:
11-03Historical (number theory)
11R37Class field theory for global fields
01A55Mathematics in the 19th century