By Peter Roquette
The twentieth century used to be a time of significant upheaval and nice growth in arithmetic. that allows you to get the final photograph of tendencies, advancements, and effects, it's illuminating to envision their manifestations in the community, within the own lives and paintings of mathematicians who have been lively in this time. The college information of Göttingen harbor a wealth of papers, letters, and manuscripts from a number of generations of mathematicians--documents which inform the tale of the old advancements from an area viewpoint. This booklet deals a couple of essays in keeping with files from Göttingen and elsewhere--essays that have no longer but been incorporated within the author's accumulated works. those essays, self sufficient from one another, are intended as contributions to the implementing mosaic of the heritage of quantity thought. they're written for mathematicians, yet there aren't any certain history necessities. 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 e-book of the eu Mathematical Society (EMS). dispensed in the Americas via the yankee Mathematical Society.
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Additional resources for Contributions to the History of Number Theory in the 20th Century
But three years later in [Alb38a] he showed that Hasse’s idea could not be realized. He presented an example of a non-cyclic division algebra of index 4 containing a pure subfield of degree 4. The base field K is the rational function field in three variables over a formally real field. 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.
The Existence Theorem can be weakened by removing the requirement for LjK to have a fixed degree n. In this weak form one is looking for a cyclic extension LjK whose degree is not specified, with the only condition that its local degrees ŒLp W Kp should be divisible by the given numbers mp (for p 2 S ). Already in 1932 Hasse had given a relatively elementary proof of this weak form [Has33a]. This does not yield the full Main Theorem but only its weak form that every central simple algebra A is similar to a cyclic algebra (which a priori does not necessarily imply A itself to be cyclic).
Algebras with pure maximal subfields. In 1934 there appeared a paper by Albert with the title “Kummer fields” [Alb34]. There, Albert proved the following theorem: A central division algebra D of prime degree p over a field K of characteristic 0 is cyclic if and only if D contains an element x … K such that x p 2 K. x/ would be a cyclic subfield of D of degree p. If K does not contain the p-thproots of unityp then Albert constructs a cyclic field LjK contained in D such that L. x; p 1/; this can be done by the classical methods of Kummer.
Contributions to the History of Number Theory in the 20th Century by Peter Roquette