By Peter Roquette

The twentieth century was once a time of serious upheaval and nice development in arithmetic. with a purpose to get the general photograph of developments, advancements, and effects, it really is illuminating to check their manifestations in the community, within the own lives and paintings of mathematicians who have been lively in this time. The college records of Göttingen harbor a wealth of papers, letters, and manuscripts from a number of generations of mathematicians--documents which inform the tale of the ancient advancements from an area perspective. This booklet deals a couple of essays in response to files from Göttingen and elsewhere--essays that have now not but been incorporated within the author's gathered works. those essays, self sustaining from one another, are intended as contributions to the implementing mosaic of the heritage of quantity conception. they're written for mathematicians, yet there are not any detailed heritage standards. 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 book of the eu Mathematical Society (EMS). disbursed in the Americas via the yank Mathematical Society.

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1 The Brauer–Hasse–Noether Theorem 35 Theorem this homomorphism is surjective. n/ . a/ Á 0 mod n. n/ ; '; a/. n/ jKp / is isomorphic to Z=n. n/ jKp / contains all central simple algebras Ap of index dividing n. n/ jKp / is split by Lp . Since the index of Ap divides n D ŒLp W Kp this follows from the Local Splitting Criterion. Thus indeed, Hasse could have answered Noether’s question with “yes”, already in 1930 ; in fact he did so later. But Noether’s conclusion that one could derive local class field theory from this, was too optimistic.

Those which are not integrally closed, carry a more complicated ideal theory.

This theorem is much stronger than Hasse’s Existence Theorem: Grunwald’s theorem. Let K be an algebraic number field and S a finite set of primes of K. For each p 2 S let there be given a cyclic field extension Lp jKp . Moreover, let n 2 N be a common multiple of the degrees ŒLp W Kp . Then there exists a cyclic field extension LjK of degree n such that for each p 2 S its completion coincides with the given fields Lp . Whereas Hasse needed only the fact that the local degrees ŒLp W Kp should be multiples of the given numbers mp (for p 2 S), Grunwald’s theorem claims that even the local fields Lp themselves can be prescribed as cyclic extensions of degree mp of Kp (for the finitely many primes p 2 S).