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I read a book "a history of abstract algebra"- chapter 6 by Israel Kleiner. And in this book, it is said that Emmy Noether gave a presentation at the ICM congress held in Zurich in 1932, and that the content was the starting point of modern cohomology theory. And the reference for this part is M.K. Smith's "Emmy Noether's contributions to mathematics, Unpublished notes (13pp, CA 1976)", but no matter how much I search on Google, I can't find a resource where I can read this paper. So in the end, what I want to ask are

  1. What did Emmy Noether present at the 1932 ICM convention?

  2. Which of the contents presented by Emmy Noether could have been the starting point of modern cohomology?

user1274233
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1 Answers1

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All past ICM proceedings are available online in one place: https://www.mathunion.org/icm/proceedings. Scroll down to the year 1932, where pp. 189-194 in volume 1 is Noether's paper, which starts on p. 197 in that pdf file. The basic point is that the multiplicative relations among the standard basis elements in a crossed product algebra are an example of a 2-cocycle. This is entirely in the direction of abstract algebra and number theory, not topology (one of your tags), so you be disappointed.

See also the section "Class field theory and cohomology" on pp. 42-45 in Chapter 1 of Peter Roquette's book Contributions to the History of Number Theory in the 20th Century. This is also available as manuscript 29 The Brauer-Hasse-Noether theorem in historical perspective on Roquette's webpage https://www.mathi.uni-heidelberg.de/~roquette/manu.html.

KCd
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    Hello, thank you so much for your really great answer. However, I have a few questions about the links you provided in your answer. Can I ask a few more questions? This part appears in Noether's 1932 paper. "Gauss zurückgehenden Fragestellungen, dem Hauptgeschlechtssatz und dem eng damit verbundenen Normensatz." Using Google Translator, this is translated into English as follows. "Questions going back to Gauss, the main gender theorem and the narrow associated set of standards." My question here is what Gauss's two theorems are that led Noether to write this paper. ​ – user1274233 Mar 06 '24 at 13:40
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    That translation is total garbage. The term Hauptgeschlechtssatz means "principal ideal theorem" (not "main gender theorem" ), which you can determine by googling that word on its own (e.g., see it in https://arxiv.org/pdf/math/0207306.pdf) and "Normensatz" means "norm theorem" (not "set of standards") and more precisely she has in mind what is called Hasse's norm theorem. Hasse had proved it in 1931. They are both part of class field theory in the 20th century, not due to Gauss. But like much of number theory, hints of these future results can presumably be traced back to work of Gauss. – KCd Mar 06 '24 at 14:31
  • Thank you for your comment!! – user1274233 Mar 06 '24 at 22:14
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    A better way to translate Hauptgeschlechtssatz would be principal genus theorem. Gauss introduced the concept of the genus of a quadratic form. The terms genus and gender can occur as the same word in some languages besides German too. – KCd Mar 07 '24 at 03:51
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    Thank you for the additional explanation. I will study by referring to the comments. – user1274233 Mar 07 '24 at 06:27