Rumely's local global principle for algebraic P<IMG ALIGN=BOTTOM >C fields over rings期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

Rumely's local global principle for algebraic PC fields over rings

Authors:	Moshe Jarden Aharon Razon

Institution:	School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel ; School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Abstract:	Let $\mathcal{S}$ be a finite set of rational primes. We denote the maximal Galois extension of $\mathbb{Q}$ in which all $p\in \mathcal{S}$ totally decompose by . We also denote the fixed field in of elements $\sigma _{1},\ldots , \sigma _{e}$ in the absolute Galois group $G( \mathbb{Q})$ of $\mathbb{Q}$ by $N( {\boldsymbol \sigma })$ . We denote the ring of integers of a given algebraic extension of $\mathbb{Q}$ by $\mathbb{Z}_{M}$ . We also denote the set of all valuations of (resp., which lie over ) by $\mathcal{V}_{M}$ (resp., $\mathcal{S}_{M}$ ). If $v\in \mathcal{V}_{M}$ , then $O_{M,v}$ denotes the ring of integers of a Henselization of with respect to . We prove that for almost all ${\boldsymbol \sigma }\in G( \mathbb{Q})^{e}$ , the field $M=N( {\boldsymbol \sigma })$ satisfies the following local global principle: Let be an affine absolutely irreducible variety defined over . Suppose that $V(O_{M,v})\not =\varnothing$ for each $v\in \mathcal{V}_{M}\backslash \mathcal{S}_{M}$ and $V_{\mathrm{sim}}(O_{M,v})\not =\varnothing$ for each $v\in \mathcal{S}_{M}$ . Then $V(O_{M})\not =\varnothing$ . We also prove two approximation theorems for .

Keywords:	PAC field over rings P$\mathcal{S}$C fields over rings local global principle global fields absolute Galois group Haar measure valuations Henselian fields field of totally $\mathcal{S}$-adic numbers

	点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Transactions of the American Mathematical Society》下载免费的PDF全文

设为首页 | 免责声明 | 关于勤云 | 加入收藏