A NOTE ON MIKI'S GENERALIZATION OF THE GRUNWALD-HASSE-WANG THEOREM

The Theorem of Grunwald -

A GENERALIZATION OF A JACOBSON’S COMMUTATIVITY THEOREM

In this paper we study the structure and the commutativity of a ring R, in which for each x,y ? R, there exist two integers depending on x,y such that [x,y]k equals x n or y n.

The theorem of Grunwald-Wang in the setting of valuation theory

Given a field K with finitely many valuations; does there exist an extension of K which at these valuations has a prescribed local behavior? The Grunwald-Wang theorem answers this question in the case of abelian field extensions. Originally developed for algebraic number fields in the context of class field theory, it has turned out that it is valid quite generally, for arbitrary multi-valued f...

Grunwald–wang for Global Character Groups

Let F denote C or Q` for a prime ` 6= char(k), equipped with its natural topology, and let Gk = Gal(ks/k) for a choice of separable closure ks/k. For each place v of k choose a separable closure kv,s/kv and an embedding ks ↪→ kv,s over k ↪→ kv, thereby identifying Gkv with a decomposition group at v in Gk. It is surprising (but confirmed by consultation with experts) that the following natural ...

The Hasse–Minkowski Theorem

The Hasse-Minkowski Theorem provides a characterization of the rational quadratic forms. What follows is a proof of the Hasse-Minkowski Theorem paraphrased from the book, Number Theory by Z.I. Borevich and I.R. Shafarevich [1]. Throughout this paper, some familiarity with the p-adic numbers and the Hilbert symbol is assumed and some basic facts about quadratic forms are stated without proof. Al...