Peter Roquette's Contributions to the History of Number Theory in the 20th PDF

By Peter Roquette

The 20 th century used to be a time of significant upheaval and nice growth in arithmetic. as a way to get the general photo of traits, advancements, and effects, it really is illuminating to check their manifestations in the neighborhood, within the own lives and paintings of mathematicians who have been energetic in this time. The collage data of Göttingen harbor a wealth of papers, letters, and manuscripts from numerous generations of mathematicians--documents which inform the tale of the ancient advancements from a neighborhood viewpoint. This ebook bargains a couple of essays in response to records from Göttingen and elsewhere--essays that have now not but been incorporated within the author's gathered works. those essays, self reliant from one another, are intended as contributions to the implementing mosaic of the heritage of quantity idea. they're written for mathematicians, yet there are not any detailed historical past requisites. The essays talk about the works of Abraham Adrian Albert, Cahit Arf, Emil Artin, Richard Brauer, Otto Grün, Helmut Hasse, Klaus Hoechsmann, Robert Langlands, Heinrich-Wolfgang Leopoldt, Emmy Noether, Abraham Robinson, Ernst Steinitz, Hermann Weyl, and others. A booklet of the eu Mathematical Society (EMS). disbursed in the Americas via the yank Mathematical Society.

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Exponent = Index. One of the important consequences of the Main Theorem is the fact that over number fields, the exponent of a central simple algebra equals its index. This is a very remarkable theorem. It has interesting consequences in the representation theory of finite groups, and this was the reason why Richard Brauer was particularly interested in it. 2 (p. 13). See also Albert [Alb32b] where a similar question is studied. 1, p. 22). To this end the number n in 35 In Hasse’s mathematical diary, dated February 1935 we find the following entry: “Proof of a theorem of Albert, following E.

To this end the number n in 35 In Hasse’s mathematical diary, dated February 1935 we find the following entry: “Proof of a theorem of Albert, following E. ” Witt’s proof of Albert’s theorem is particularly simple, following the style of Emmy Noether, but the essential ingredients are the same as in Albert’s proof. It seems that Witt had presented this proof in the Hasse seminar, and that Hasse had noted it in his diary for future reference. 36 From the references in those papers it appears that Brauer and Albert did not know the results of each other concerning this question.

Kp / ! 2 0 if p is complex. K/ for a number field K. If A is a central simple algebra over A K and Ap its p-adic completion then Hasse writes briefly A instead of pp . K/ ! ˚ Q=Z p where the prime on the † sign of the direct sum should remind the reader that if p is infinite then Q=Z has to be replaced by 12 Z=Z or 0 according to whether p is real or complex. By the Local-Global Principle this global invariant map is injective. K/ is uniquely determined by its Hasse invariants (up to similarity).

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