# Two implicit references in Serre's *Groupes de Galois : le cas abélien*

In his exposé at the Galois bicentenary conference, Serre makes two references which are not quite explicit.

The first reference occurs (at 22:30 in the video) when he is talking about Dedekind's XIth supplement to Dirichlet's Vorlesungen and says that a certain French graduate text on number fields is essentially based on that supplement.

Question Which book is he referring to ?

I think it is Pierre Samuel's Théorie algébrique des nombres, but would like to hear other people's opinions.

The second reference occurs (at 47:00 in the video) when he is discussing Artin's general reciprocity law and says that its formulation was so simple that apparently his contemporaries did not quite believe it.

Question Which arithmeticians does Serre have in mind ?

I believe he is thinking of Hasse in particular. Artin is reported to have said that Hasse told him that the conjectural general reciprocity law couldn't possibly be true.

Which other mathematicians of the time expressed their disbelief (before Artin actually proved his own conjecture) ?

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Can't you ask him? –  Mariano Suárez-Alvarez Jan 15 '12 at 5:27
If I meet him, I would. In the meantime, it would be interesting to hear from MOers who know the history of number theory in the 20s. –  Chandan Singh Dalawat Jan 15 '12 at 6:15
After reading just the questions, I immediately thought "Samuel" and "Hasse". As for the second, I do not think that the plural is justified: Artin published his reciprocity law in 1923, and at that time, only a couple of mathematicians had studied Takagi's work. The ones I can think of are Artin, Hasse, Hecke, Siegel, Fueter, Furtwängler. –  Franz Lemmermeyer Jan 15 '12 at 9:51
Franz, Mattuck quotes Artin as saying that Then I showed it to the other number theorists, but they all laughed at it, and I remember Hasse in particular telling me it couldn't possibly be true. –  Chandan Singh Dalawat Jan 15 '12 at 11:53

I certainly don't claim that I can answer your questions authoritatively , but here are two small remarks.

1) Samuel's book is certainly an excellent guess: it is actually the only textbook in French I can think of entirely devoted to elementary algebraic number theory.

2) There was a preliminary draft of a text by Bourbaki to be inserted in His Commutative Algebra, called Contre-Rédaction de la Différente, rédaction n°410.
It was written by Samuel, and Bourbaki's étiquette was that if one of His collaborators had written a text that was not to be integrated in the corresponding volume in a foreseeable future, then that person was free to publish it on his own.

Precedents include Lang's book on the cohomology of groups, and Godement's classic on sheaf theory.
Acknowledgment would be given in some coded way like Godement's amusing:
Il est bien évident que ce livre n'aurait jamais vu le jour sans l'aide précieuse et les encouragements enthousiastes (quoique partiellement intéressés) que nous ont prodigué [sic] certains géomètres et tout spécialement N.Bourbaki ...

This applies to the case at hand: Samuel's book is quite similar to Bourbaki's draft, with obvious changes stemming from the fact that of course Samuel couldn't use sophisticated tools, like étale algebras, used in the draft.
Samuel writes ...I want particularly to thank the master of my generation,N.Bourbaki,who has had the kindness to show me his unpublished manuscripts...

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That's interesting. I was under the impression that the only Bourbaki draft touching upon number theory was (R171_nbr_072) Rapport d'algèbre unidimensionnelle. Chap. II Arithmétique des corps de nombres algébriques (73 p.) (mathdoc.emath.fr/archives-bourbaki/PDF/171_nbr_072.pdf) I couldn't find Contre-Rédaction de la Différente, rédaction n°410 on this site. –  Chandan Singh Dalawat Jan 15 '12 at 11:45
"His Commutative Algebra" :D –  Gunnar Magnusson Jan 15 '12 at 11:55
Dear @Chandan: there are many texts which are not (yet ?) on the site. I own n°410 (and a few others) on movingly yellowing paper, courtesy of several collaborators that Bourbaki has had in Nice (starting of course with Dieudonné who created our department) –  Georges Elencwajg Jan 15 '12 at 12:09
Quelle chance ! –  Chandan Singh Dalawat Jan 15 '12 at 12:11