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By Tsvetkov V. M.

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Thus a close connection between the theory of algebras and class field theory became visible. 1 below. 3 Remarks 45 While Artin’s paper [Art:1927] with his reciprocity law had been named as the “Coronation of class field theory”, similarly Hasse’s paper [Has:1933] could now be regarded as the “Coronation of the theory of algebras”. We can now understand Artin’s exclamation which we have cited in Section 2 , namely that he regards this as the “greatest advance in Number Theory of the last years. .

Grunwald’s thesis does not contain the Existence Theorem, but Hasse discovered that Grunwald’s methods could be used to obtain a proof of the theorem. From the correspondence between Grunwald and Hasse (which is preserved) we can infer that Hasse had proposed to Grunwald to extract from his thesis a proof of the Existence Theorem and publish it in a separate paper. And Grunwald did so. The reference “Grunwald [2]” in Hasse’s [Has:1933] refers to Grunwald’s paper, at that time still “forthcoming”, which appeared 1933 in Crelle’s Journal [Gru:1933].

32 5. From the Local-Global Principle to the Main Theorem It is understood that representations are to be meant over a field of characteristic 0. 32 Two representations over the same field are equivalent if they can be transformed into each other by a non-singular matrix; in “modern” terms: if they determine isomorphic G-modules. This theorem had been conjectured by I. Schur [Schu:1906] but with h = 1. In view of this Hasse did not say that his theorem is a proof of Schur’s conjecture; instead he says that this theorem constitutes a√ “support” of Schur’s conjecture.

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