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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201201.png" /> be a global field. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201202.png" /> is either a number field, i.e. a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201203.png" />, or a function field of one variable over a [[Finite field|finite field]]. Denote the algebraic (respectively, separable) closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201204.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201205.png" /> (respectively, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201206.png" />; cf. also [[Extension of a field|Extension of a field]]). A prime divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201207.png" /> is an equivalence class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201208.png" /> of absolute values (cf. also [[Norm on a field|Norm on a field]]). For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l1201209.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012010.png" /> be a representative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012011.png" />. Denote the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012013.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012014.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012015.png" /> is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012016.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012017.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012018.png" /> is metric), or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012019.png" /> is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012020.png" /> or a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012022.png" /> is ultra-metric).
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There is a natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012023.png" />-topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012024.png" /> whose basic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012025.png" />-open subsets have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012026.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012029.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012030.png" />-topology has compatible extensions to all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012032.png" /> is an affine algebraic set over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012033.png" />. In each case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012034.png" /> is locally compact.
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Embed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012035.png" /> into the [[Algebraic closure|algebraic closure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012036.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012037.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012038.png" /> is a real (respectively, algebraic) closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012039.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012040.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012041.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012042.png" />), and is a Henselization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012043.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012044.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012045.png" /> is ultra-metric (cf. also [[Henselization of a valued field|Henselization of a valued field]]). In the latter case, the valuation ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012046.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012047.png" />. In each case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012048.png" /> is uniquely determined up to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012049.png" />-isomorphism.
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Let $K$ be a global field. In other words, $K$ is either a number field, i.e. a finite extension of $\mathbf{Q}$, or a function field of one variable over a [[Finite field|finite field]]. Denote the algebraic (respectively, separable) closure of $K$ by $\widetilde { K }$ (respectively, by $K _ { s }$; cf. also [[Extension of a field|Extension of a field]]). A prime divisor of $K$ is an equivalence class $\text{p}$ of absolute values (cf. also [[Norm on a field|Norm on a field]]). For each $\text{p}$, let $|.|_{\operatorname{p}}$ be a representative of $\text{p}$. Denote the completion of $K$ at $\text{p}$ by $\hat { K } _ { \text{p} }$. Then $\hat { K } _ { \text{p} }$ is either $\mathbf{R}$ or $\mathbf{C}$ ($\text{p}$ is metric), or $\hat { K } _ { \text{p} }$ is a finite extension of $\hat { \mathbf{Q} } _ { p }$ or a finite extension of $\mathbf{F} _ { p } ( ( t ) )$ ($\text{p}$ is ultra-metric).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012050.png" /> is a set of prime divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012052.png" /> is an algebraic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012053.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012054.png" /> denotes the set of all extensions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012055.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012056.png" />.
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There is a natural $\text{p}$-topology on $\hat { K } _ { \text{p} }$ whose basic $\text{p}$-open subsets have the form $\left\{ x \in \widehat { K } _ { \operatorname {p} } : | x - a | _ { \operatorname {p} } \leq \epsilon \right\}$, for $a \in \hat { K } _ { \text{p} }$ and $\epsilon \in \mathbf{R}$, $\epsilon &gt; 0$. The $\text{p}$-topology has compatible extensions to all sets $V ( \widehat { K } _ { \operatorname {p} } )$, where $V$ is an affine algebraic set over $\hat { K } _ { \text{p} }$. In each case, $V ( \widehat { K } _ { \operatorname {p} } )$ is locally compact.
  
In the sequel, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012057.png" /> be a fixed set of prime divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012058.png" /> which does not contain all prime divisors. For each algebraic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012059.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012060.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012061.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012062.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012063.png" /> is metric, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012064.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012065.png" />-unit ball and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012066.png" /> is ultra-metric, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012067.png" /> is the valuation ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012068.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012069.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012070.png" /> consists of ultra-metric primes only, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012071.png" /> is a Dedekind domain (cf. also [[Dedekind ring|Dedekind ring]]). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012073.png" /> consists of all prime numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012074.png" />.
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Embed $\widetilde { K }$ into the [[Algebraic closure|algebraic closure]] of $\hat { K } _ { \text{p} }$ and let $K _ { \text{p} } = K _ { s } \cap \hat { K } _ { \text{p} }$. Then $K _ {  \operatorname{p} }$ is a real (respectively, algebraic) closure of $K$ at $\text{p}$ if $\hat { K } _ { \text{p} } = \bf R$ (respectively, $\hat { K } _ { \operatorname{p} } = \mathbf{C}$), and is a Henselization of $K$ at $\text{p}$ if $\text{p}$ is ultra-metric (cf. also [[Henselization of a valued field|Henselization of a valued field]]). In the latter case, the valuation ring of $K _ {  \operatorname{p} }$ is denoted by $O _ { K , \text{p} }$. In each case, $K _ {  \operatorname{p} }$ is uniquely determined up to a $K$-isomorphism.
  
Fix also a finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012075.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012076.png" />. Consider the field of totally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012078.png" />-adic numbers:
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If $P$ is a set of prime divisors of $K$ and $L$ is an algebraic extension of $K$, then $P _ { L }$ denotes the set of all extensions to $L$ of all $\operatorname{p} \in P$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012079.png" /></td> </tr></table>
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In the sequel, let $P$ be a fixed set of prime divisors of $K$ which does not contain all prime divisors. For each algebraic extension $L$ of $K$ and each $\operatorname { p} \in P _ { L }$, let $O _ { \text{p} } = \{ x \in L : | x | _ { \text{p} } \leq 1 \}$. Thus, if $\text{p}$ is metric, then $O _ { \text{p} }$ is the $\text{p}$-unit ball and if $\text{p}$ is ultra-metric, then $O _ { \text{p} }$ is the valuation ring of $\text{p}$. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012069.png"/>. If $P$ consists of ultra-metric primes only, then $O _ { K }$ is a Dedekind domain (cf. also [[Dedekind ring|Dedekind ring]]). For example, if $K = \mathbf{Q}$ and $P$ consists of all prime numbers, then $O _ { K } = \mathbf{Z}$.
  
This is the largest Galois extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012080.png" /> in which each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012081.png" /> totally splits. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012085.png" /> is empty, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012087.png" /> is the ring of all algebraic integers. The following two theorems, which can be found in [[#References|[a10]]] and [[#References|[a4]]], are therefore generalizations of Rumely's local-global principle and the density theorem (cf. also [[Local-global principles for the ring of algebraic integers|Local-global principles for the ring of algebraic integers]]).
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Fix also a finite subset $S$ of $P$. Consider the field of totally $\text{p}$-adic numbers:
  
The local-global principle: In the above notation, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012088.png" />. Consider an absolutely irreducible affine variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012089.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012090.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012091.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012092.png" />. Suppose further that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012093.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012094.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012095.png" />.
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\begin{equation*} K _ { \operatorname{tot} S } = \bigcap _ { \operatorname{p} \in S } \bigcap _ { \sigma \in G ( K ) } K _ { \operatorname{p} } ^ { \sigma } \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012096.png" /> is the Zariski-open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012097.png" /> consisting of all non-singular points.
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This is the largest Galois extension of $K$ in which each $\text{p} \in S$ totally splits. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012082.png"/>. If $K = \mathbf{Q}$, $O _ { K } = \mathbf{Z}$ and $S$ is empty, then $K _ { \text { tot }S } = \tilde { \mathbf{Q} }$ and $\mathbf{Z} _ { \text { tot } S } = \tilde{\mathbf{Z}}$ is the ring of all algebraic integers. The following two theorems, which can be found in [[#References|[a10]]] and [[#References|[a4]]], are therefore generalizations of Rumely's local-global principle and the density theorem (cf. also [[Local-global principles for the ring of algebraic integers|Local-global principles for the ring of algebraic integers]]).
  
The density theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l12012099.png" /> be as in the local-global principle. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120100.png" /> be a finite subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120101.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120102.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120103.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120104.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120105.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120106.png" /> be a non-empty <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120107.png" />-open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120108.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120109.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120110.png" /> be a non-empty <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120111.png" />-open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120112.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120113.png" /> contains a point which lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120114.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120115.png" />.
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The local-global principle: In the above notation, let $M = K_{totS }$. Consider an absolutely irreducible affine variety $V$ over $K$. Suppose that $V ( O _ { K , \text{p} } ) \neq \emptyset$ for each $\operatorname{p} \in P$. Suppose further that $V _ { \text { simp } } ( O _ { K , p } ) \neq \emptyset$ for each $\text{p} \in S$. Then $V ( O _ { M } ) \neq \emptyset$.
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Here, $V _ { \text { simp } }$ is the Zariski-open subset of $V$ consisting of all non-singular points.
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The density theorem: Let $M$ and $V$ be as in the local-global principle. Let $T$ be a finite subset of $P$ containing $S$. Suppose that $V ( O _ { K , \text{p} } ) \neq \emptyset$ for each $\operatorname{p} \in P$. For each $\text{p} \in S$, let ${\cal U} _ { \operatorname { p } }$ be a non-empty $\text{p}$-open subset of $V _ { \text { simp } } ( K _ { p } )$. For each $\text{p} \in T \backslash S$, let ${\cal U} _ { \operatorname { p } }$ be a non-empty $\text{p}$-open subset of $V ( K _ { \text{p} } )$. Then $V ( O _ { M } )$ contains a point which lies in ${\cal U} _ { \operatorname { p } }$ for each $\operatorname{p} \in T$.
  
 
Although the density theorem looks stronger than the local-global principle, one can actually use the weak approximation theorem and deduce the density theorem from the local-global principle.
 
Although the density theorem looks stronger than the local-global principle, one can actually use the weak approximation theorem and deduce the density theorem from the local-global principle.
  
Both the local-global principle and the density theorem are actually true for fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120116.png" /> which are much smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120117.png" />. To this end, call a field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120118.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120119.png" /> PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120121.png" /> if for every dominating separable rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120122.png" /> of absolutely irreducible varieties of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120123.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120124.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120125.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120126.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120127.png" /> is a number field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120128.png" /> consists of ultra-metric primes only, [[#References|[a8]]], Thm. 1.4; 1.5, imply both the density theorem and the local-global principle for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120129.png" />. In the function field case, [[#References|[a8]]] must replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120130.png" /> by its maximal [[purely inseparable extension]], which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120131.png" />. Accordingly, the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120132.png" /> in the assumption of the density theorem and the local-global principle must be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120133.png" />. However, using the methods of [[#References|[a4]]] and [[#References|[a5]]], it is plausible that even in this case one can restore the theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120134.png" />.
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Both the local-global principle and the density theorem are actually true for fields $M$ which are much smaller than $K _ { \text{tot }S }$. To this end, call a field extension $M^{\prime}$ of $K$ [[PAC]] over $O _ { K }$ if for every dominating separable rational mapping $\phi : V \rightarrow \mathbf A ^ { r }$ of absolutely irreducible varieties of dimension $r$ over $M^{\prime}$, there exists an $x \in V ( M ^ { \prime } )$ such that $\phi ( x ) \in O^r_K$. If $K$ is a number field and $P$ consists of ultra-metric primes only, [[#References|[a8]]], Thm. 1.4; 1.5, imply both the density theorem and the local-global principle for $M = M ^ { \prime } \cap K _ { \operatorname { tot } S }$. In the function field case, [[#References|[a8]]] must replace $M$ by its maximal [[purely inseparable extension]], which is denoted by $M _ { \operatorname {ins} }$. Accordingly, the fields $K _ {  \operatorname{p} }$ in the assumption of the density theorem and the local-global principle must be replaced by $( K _ { p } ) _ { \text{ins} }$. However, using the methods of [[#References|[a4]]] and [[#References|[a5]]], it is plausible that even in this case one can restore the theorem for $M = M ^ { \prime } \cap K _ { \operatorname { tot } S }$.
  
By Hilbert's Nullstellensatz (cf. also [[Hilbert theorem|Hilbert theorem]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120135.png" /> is PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120136.png" />. Hence, [[#References|[a8]]], Thms.1.4; 1.5, generalize the density theorem and the local-global principle above. Probability theory supplies an abundance of other algebraic extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120137.png" /> which are PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120138.png" />. The measure space in question is the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120139.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120140.png" /> copies of the absolute [[Galois group|Galois group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120141.png" /> equipped with the [[Haar measure|Haar measure]]. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120142.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120143.png" /> be the fixed field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120144.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120145.png" />. By [[#References|[a7]]], Prop. 3.1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120146.png" /> is PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120147.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120148.png" />. Together with the preceding paragraph, this yields the following result (the Jarden–Razon theorem): For every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120149.png" /> and for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120150.png" />, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120151.png" /> satisfies the conclusions of the local-global principle and the density theorem.
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By Hilbert's Nullstellensatz (cf. also [[Hilbert theorem|Hilbert theorem]]), $K _ { s }$ is PAC over $O _ { K }$. Hence, [[#References|[a8]]], Thms.1.4; 1.5, generalize the density theorem and the local-global principle above. Probability theory supplies an abundance of other algebraic extensions of $K$ which are PAC over $O _ { K }$. The measure space in question is the Cartesian product $G ( K ) ^ { e }$ of $e$ copies of the absolute [[Galois group|Galois group]] of $K$ equipped with the [[Haar measure|Haar measure]]. For each $\overline { \sigma } = ( \sigma _ { 1 } , \ldots , \sigma _ { e } ) \in G ( K ) ^ { e }$, let $K _ { s } ( \overline { \sigma } )$ be the fixed field of $\sigma _ { 1 } , \ldots , \sigma _ { e }$ in $K _ { s }$. By [[#References|[a7]]], Prop. 3.1, $K _ { s } ( \overline { \sigma } )$ is PAC over $O _ { K }$ for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. Together with the preceding paragraph, this yields the following result (the Jarden–Razon theorem): For every positive integer $e$ and for almost all $\overline { \sigma } \in G ( K ) ^ { e }$, the field $M = ( K _ { s } ( \overline { \sigma } ) \cap K _ { \text{tot } S} ) _ { \text{in s} }$ satisfies the conclusions of the local-global principle and the density theorem.
  
The local-global principle for rings implies a local-global principle for fields. An algebraic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120152.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120153.png" /> is said to be P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120155.png" />C (pseudo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120157.png" />-adically closed) if each absolutely irreducible variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120158.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120159.png" /> which has a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120160.png" />-rational point for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120161.png" />, has an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120162.png" />-rational point. In particular, by the local-global principle and the Jarden–Razon theorem, the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120164.png" /> are P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120165.png" />C for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120166.png" />. The main result of [[#References|[a5]]] supplies P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120167.png" />C extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120168.png" /> which are even smaller than the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120169.png" /> (the Geyer–Jarden theorem): For every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120170.png" /> and for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120171.png" />, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120172.png" /> is P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120173.png" />C.
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The local-global principle for rings implies a local-global principle for fields. An algebraic extension $M$ of $K$ is said to be P$S$C (pseudo $S$-adically closed) if each absolutely irreducible variety $V$ over $M$ which has a simple $M _ { \operatorname{p} }$-rational point for each $\operatorname{p} \in S _ { M}$, has an $M$-rational point. In particular, by the local-global principle and the Jarden–Razon theorem, the fields $K _ { \text{tot }S }$ and $( K _ { s } ( \overline { \sigma } ) \cap K _ { totS }  ) _ { ins }$ are P$S$C for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. The main result of [[#References|[a5]]] supplies P$S$C extensions of $K$ which are even smaller than the fields $K _ { s } ( \overline { \sigma } ) \cap K _ { \operatorname{tot}S }$ (the Geyer–Jarden theorem): For every positive integer $e$ and for almost all $\overline { \sigma } \in G ( K ) ^ { e }$, the field $K _ { s } [ \overline { \sigma } ] \cap K _ { \text{tot }S }$ is P$S$C.
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120174.png" /> is the maximal Galois extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120175.png" /> that is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120176.png" />. It is not known (1998) whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120177.png" /> satisfies the local-global principle. (So, the Geyer–Jarden theorem is not a consequence of the Jarden–Razon theorem.) Since a separable algebraic extension of a P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120178.png" />C field is P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120179.png" />C [[#References|[a9]]], Lemma 7.2, the Geyer–Jarden theorem implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120180.png" /> is P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120181.png" />C for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120182.png" />. Likewise, it reproves that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120183.png" /> is P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120184.png" />C.
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Here, $K _ { S } [ \overline { \sigma } ]$ is the maximal Galois extension of $K$ that is contained in $K _ { s } ( \overline { \sigma } )$. It is not known (1998) whether $O _ { K _ { s } [ \bar{\sigma} ] } $ satisfies the local-global principle. (So, the Geyer–Jarden theorem is not a consequence of the Jarden–Razon theorem.) Since a separable algebraic extension of a P$S$C field is P$S$C [[#References|[a9]]], Lemma 7.2, the Geyer–Jarden theorem implies that $K _ { s } ( \overline { \sigma } ) \cap K _ { \operatorname{tot}S }$ is P$S$C for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. Likewise, it reproves that $K _ { \text{tot }S }$ is P$S$C.
  
A field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120185.png" /> which is P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120186.png" />C is also ample (i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120187.png" /> is an absolutely irreducible variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120188.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120189.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120190.png" /> is Zariski-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120191.png" />). Ample fields, in particular P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120192.png" />C fields, have the nice property that the inverse problem of Galois theory over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120193.png" /> has a positive solution (cf. also [[Galois theory, inverse problem of|Galois theory, inverse problem of]]). That is, for every finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120194.png" /> there exists a Galois extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120195.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120196.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120197.png" />. Indeed, every finite split embedding problem over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120198.png" /> is solvable [[#References|[a11]]], Main Thm. A, [[#References|[a6]]], Thm. 2.
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A field $M$ which is P$S$C is also ample (i.e. if $V$ is an absolutely irreducible variety over $M$ and $V _ { \text { simp } } ( M ) \neq \emptyset$, then $V ( M )$ is Zariski-dense in $V$). Ample fields, in particular P$S$C fields, have the nice property that the inverse problem of Galois theory over $M ( t )$ has a positive solution (cf. also [[Galois theory, inverse problem of|Galois theory, inverse problem of]]). That is, for every finite group $G$ there exists a Galois extension $F$ of $M ( t )$ such that $\operatorname{Gal}( F / M ( t ) ) \cong G$. Indeed, every finite split embedding problem over $M ( t )$ is solvable [[#References|[a11]]], Main Thm. A, [[#References|[a6]]], Thm. 2.
  
Another interesting consequence of the local-global principle describes the absolute Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120199.png" />: It is due to F. Pop [[#References|[a11]]], Thm. 3, and may be considered as a local-global principle for the absolute Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120200.png" /> (Pop's theorem): The absolute Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120201.png" /> is the free pro-finite product
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Another interesting consequence of the local-global principle describes the absolute Galois group of $K _ { \text{tot }S }$: It is due to F. Pop [[#References|[a11]]], Thm. 3, and may be considered as a local-global principle for the absolute Galois group of $K _ { \text{tot }S }$ (Pop's theorem): The absolute Galois group of $K _ { \text{tot }S }$ is the free pro-finite product
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120202.png" /></td> </tr></table>
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\begin{equation*} \prod _ { \text{p} ^ { \prime } \in S ^ { \prime } } G ( K _ { \text{p}  ^ { \prime } } ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120203.png" /> is the set of all extensions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120204.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120205.png" />. This means that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120206.png" /> is a finite group, then each continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120207.png" /> whose restriction to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120208.png" /> is a homomorphism, can be uniquely extended to a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120209.png" />.
+
where $S ^ { \prime }$ is the set of all extensions to $K _ { \text{tot }S }$ of all $\text{p} \in S$. This means that if $G$ is a finite group, then each continuous mapping $\alpha _ { 0 } : \cup _ { \text { p } ^ { \prime } \in S ^ { \prime } } G ( K _ { \text { p } ^ { \prime } } ) \rightarrow G$ whose restriction to each $G ( K _ {  \operatorname{p} ^ { \prime } } )$ is a homomorphism, can be uniquely extended to a homomorphism $\alpha : G ( K _ { \operatorname { tot } S }  ) \rightarrow G$.
  
As a consequence of the local-global principle, Yu.L. Ershov [[#References|[a2]]], Thm. 3, has proved that the [[Elementary theory|elementary theory]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120210.png" /> is decidable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120212.png" /> does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120213.png" />, this implies, by [[#References|[a1]]], p. 86; Corol. 10, that the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120214.png" /> is decidable. In particular, Hilbert's tenth problem has an affirmative solution over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120215.png" />. If however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120216.png" />, then the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120217.png" /> is decidable [[#References|[a3]]] but the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120218.png" /> is undecidable [[#References|[a12]]].
+
As a consequence of the local-global principle, Yu.L. Ershov [[#References|[a2]]], Thm. 3, has proved that the [[Elementary theory|elementary theory]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120210.png"/> is decidable. If $S$ does not contain $\infty$, this implies, by [[#References|[a1]]], p. 86; Corol. 10, that the elementary theory of ${\bf Z} _ { \text{tot} S }$ is decidable. In particular, Hilbert's tenth problem has an affirmative solution over ${\bf Z} _ { \text{tot} S }$. If however, $S = \{ \infty \}$, then the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120217.png"/> is decidable [[#References|[a3]]] but the elementary theory of ${\bf Z} _ { \text{tot} S }$ is undecidable [[#References|[a12]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Darnière,  "Étude modèle-théorique d'anneaus satisfaisant un principe de Hasse non singulier"  ''PhD Thesis''  (1998)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Yu.L. Ershov,  "Nice local-global fields I"  ''Algebra and Logic'' , '''35'''  (1996)  pp. 229–235</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.D. Fried,  D. Haran,  H. Völklein,  "Real hilbertianity and the field of totally real numbers"  ''Contemp. Math.'' , '''74'''  (1994)  pp. 1–34</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Green,  F. Pop,  P. Roquette,  "On Rumely's local-global principle"  ''Jahresber. Deutsch. Math. Ver.'' , '''97'''  (1995)  pp. 43–74</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.-D. Geyer,  M. Jarden,  "PSC Galois extensions of Hilbertian fields"  ''Manuscript Tel Aviv''  (1998)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. Haran,  M. Jarden,  "Regular split embedding problems over function fields of one variable over ample fields"  ''J. Algebra'' , '''208'''  (1998)  pp. 147–164</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Jarden,  A. Razon,  "Pseudo algebraically closed fields over rings"  ''Israel J. Math.'' , '''86'''  (1994)  pp. 25–59</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Jarden,  A. Razon,  "Rumely's local global principle for algebraic P<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120219.png" />C fields over rings"  ''Trans. Amer. Math. Soc.'' , '''350'''  (1998)  pp. 55–85</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Jarden,  "Algebraic realization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120220.png" />-adically projective groups"  ''Compositio Math.'' , '''79'''  (1991)  pp. 21–62</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  L. Moret-Bailly,  "Groupes de Picard et problèmes de Skolem II"  ''Ann. Sci. Ecole Norm. Sup.'' , '''22'''  (1989)  pp. 181–194</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  F. Pop,  "Embedding problems over large fields"  ''Ann. of Math.'' , '''144'''  (1996)  pp. 1–34</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J. Robinson,  "On the decision problem for algebraic rings" , ''Studies Math. Anal. Rel. Topics'' , Stanford Univ. Press  (1962)  pp. 297–304.</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  L. Darnière,  "Étude modèle-théorique d'anneaus satisfaisant un principe de Hasse non singulier"  ''PhD Thesis''  (1998)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  Yu.L. Ershov,  "Nice local-global fields I"  ''Algebra and Logic'' , '''35'''  (1996)  pp. 229–235</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M.D. Fried,  D. Haran,  H. Völklein,  "Real hilbertianity and the field of totally real numbers"  ''Contemp. Math.'' , '''74'''  (1994)  pp. 1–34</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  B. Green,  F. Pop,  P. Roquette,  "On Rumely's local-global principle"  ''Jahresber. Deutsch. Math. Ver.'' , '''97'''  (1995)  pp. 43–74</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  W.-D. Geyer,  M. Jarden,  "PSC Galois extensions of Hilbertian fields"  ''Manuscript Tel Aviv''  (1998)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  D. Haran,  M. Jarden,  "Regular split embedding problems over function fields of one variable over ample fields"  ''J. Algebra'' , '''208'''  (1998)  pp. 147–164</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  M. Jarden,  A. Razon,  "Pseudo algebraically closed fields over rings"  ''Israel J. Math.'' , '''86'''  (1994)  pp. 25–59</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  M. Jarden,  A. Razon,  "Rumely's local global principle for algebraic P$\mathcal{S}$C fields over rings"  ''Trans. Amer. Math. Soc.'' , '''350'''  (1998)  pp. 55–85</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  M. Jarden,  "Algebraic realization of $p$-adically projective groups"  ''Compositio Math.'' , '''79'''  (1991)  pp. 21–62</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  L. Moret-Bailly,  "Groupes de Picard et problèmes de Skolem II"  ''Ann. Sci. Ecole Norm. Sup.'' , '''22'''  (1989)  pp. 181–194</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  F. Pop,  "Embedding problems over large fields"  ''Ann. of Math.'' , '''144'''  (1996)  pp. 1–34</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  J. Robinson,  "On the decision problem for algebraic rings" , ''Studies Math. Anal. Rel. Topics'' , Stanford Univ. Press  (1962)  pp. 297–304.</td></tr></table>

Revision as of 17:43, 1 July 2020

Let $K$ be a global field. In other words, $K$ is either a number field, i.e. a finite extension of $\mathbf{Q}$, or a function field of one variable over a finite field. Denote the algebraic (respectively, separable) closure of $K$ by $\widetilde { K }$ (respectively, by $K _ { s }$; cf. also Extension of a field). A prime divisor of $K$ is an equivalence class $\text{p}$ of absolute values (cf. also Norm on a field). For each $\text{p}$, let $|.|_{\operatorname{p}}$ be a representative of $\text{p}$. Denote the completion of $K$ at $\text{p}$ by $\hat { K } _ { \text{p} }$. Then $\hat { K } _ { \text{p} }$ is either $\mathbf{R}$ or $\mathbf{C}$ ($\text{p}$ is metric), or $\hat { K } _ { \text{p} }$ is a finite extension of $\hat { \mathbf{Q} } _ { p }$ or a finite extension of $\mathbf{F} _ { p } ( ( t ) )$ ($\text{p}$ is ultra-metric).

There is a natural $\text{p}$-topology on $\hat { K } _ { \text{p} }$ whose basic $\text{p}$-open subsets have the form $\left\{ x \in \widehat { K } _ { \operatorname {p} } : | x - a | _ { \operatorname {p} } \leq \epsilon \right\}$, for $a \in \hat { K } _ { \text{p} }$ and $\epsilon \in \mathbf{R}$, $\epsilon > 0$. The $\text{p}$-topology has compatible extensions to all sets $V ( \widehat { K } _ { \operatorname {p} } )$, where $V$ is an affine algebraic set over $\hat { K } _ { \text{p} }$. In each case, $V ( \widehat { K } _ { \operatorname {p} } )$ is locally compact.

Embed $\widetilde { K }$ into the algebraic closure of $\hat { K } _ { \text{p} }$ and let $K _ { \text{p} } = K _ { s } \cap \hat { K } _ { \text{p} }$. Then $K _ { \operatorname{p} }$ is a real (respectively, algebraic) closure of $K$ at $\text{p}$ if $\hat { K } _ { \text{p} } = \bf R$ (respectively, $\hat { K } _ { \operatorname{p} } = \mathbf{C}$), and is a Henselization of $K$ at $\text{p}$ if $\text{p}$ is ultra-metric (cf. also Henselization of a valued field). In the latter case, the valuation ring of $K _ { \operatorname{p} }$ is denoted by $O _ { K , \text{p} }$. In each case, $K _ { \operatorname{p} }$ is uniquely determined up to a $K$-isomorphism.

If $P$ is a set of prime divisors of $K$ and $L$ is an algebraic extension of $K$, then $P _ { L }$ denotes the set of all extensions to $L$ of all $\operatorname{p} \in P$.

In the sequel, let $P$ be a fixed set of prime divisors of $K$ which does not contain all prime divisors. For each algebraic extension $L$ of $K$ and each $\operatorname { p} \in P _ { L }$, let $O _ { \text{p} } = \{ x \in L : | x | _ { \text{p} } \leq 1 \}$. Thus, if $\text{p}$ is metric, then $O _ { \text{p} }$ is the $\text{p}$-unit ball and if $\text{p}$ is ultra-metric, then $O _ { \text{p} }$ is the valuation ring of $\text{p}$. Let . If $P$ consists of ultra-metric primes only, then $O _ { K }$ is a Dedekind domain (cf. also Dedekind ring). For example, if $K = \mathbf{Q}$ and $P$ consists of all prime numbers, then $O _ { K } = \mathbf{Z}$.

Fix also a finite subset $S$ of $P$. Consider the field of totally $\text{p}$-adic numbers:

\begin{equation*} K _ { \operatorname{tot} S } = \bigcap _ { \operatorname{p} \in S } \bigcap _ { \sigma \in G ( K ) } K _ { \operatorname{p} } ^ { \sigma } \end{equation*}

This is the largest Galois extension of $K$ in which each $\text{p} \in S$ totally splits. Let . If $K = \mathbf{Q}$, $O _ { K } = \mathbf{Z}$ and $S$ is empty, then $K _ { \text { tot }S } = \tilde { \mathbf{Q} }$ and $\mathbf{Z} _ { \text { tot } S } = \tilde{\mathbf{Z}}$ is the ring of all algebraic integers. The following two theorems, which can be found in [a10] and [a4], are therefore generalizations of Rumely's local-global principle and the density theorem (cf. also Local-global principles for the ring of algebraic integers).

The local-global principle: In the above notation, let $M = K_{totS }$. Consider an absolutely irreducible affine variety $V$ over $K$. Suppose that $V ( O _ { K , \text{p} } ) \neq \emptyset$ for each $\operatorname{p} \in P$. Suppose further that $V _ { \text { simp } } ( O _ { K , p } ) \neq \emptyset$ for each $\text{p} \in S$. Then $V ( O _ { M } ) \neq \emptyset$.

Here, $V _ { \text { simp } }$ is the Zariski-open subset of $V$ consisting of all non-singular points.

The density theorem: Let $M$ and $V$ be as in the local-global principle. Let $T$ be a finite subset of $P$ containing $S$. Suppose that $V ( O _ { K , \text{p} } ) \neq \emptyset$ for each $\operatorname{p} \in P$. For each $\text{p} \in S$, let ${\cal U} _ { \operatorname { p } }$ be a non-empty $\text{p}$-open subset of $V _ { \text { simp } } ( K _ { p } )$. For each $\text{p} \in T \backslash S$, let ${\cal U} _ { \operatorname { p } }$ be a non-empty $\text{p}$-open subset of $V ( K _ { \text{p} } )$. Then $V ( O _ { M } )$ contains a point which lies in ${\cal U} _ { \operatorname { p } }$ for each $\operatorname{p} \in T$.

Although the density theorem looks stronger than the local-global principle, one can actually use the weak approximation theorem and deduce the density theorem from the local-global principle.

Both the local-global principle and the density theorem are actually true for fields $M$ which are much smaller than $K _ { \text{tot }S }$. To this end, call a field extension $M^{\prime}$ of $K$ PAC over $O _ { K }$ if for every dominating separable rational mapping $\phi : V \rightarrow \mathbf A ^ { r }$ of absolutely irreducible varieties of dimension $r$ over $M^{\prime}$, there exists an $x \in V ( M ^ { \prime } )$ such that $\phi ( x ) \in O^r_K$. If $K$ is a number field and $P$ consists of ultra-metric primes only, [a8], Thm. 1.4; 1.5, imply both the density theorem and the local-global principle for $M = M ^ { \prime } \cap K _ { \operatorname { tot } S }$. In the function field case, [a8] must replace $M$ by its maximal purely inseparable extension, which is denoted by $M _ { \operatorname {ins} }$. Accordingly, the fields $K _ { \operatorname{p} }$ in the assumption of the density theorem and the local-global principle must be replaced by $( K _ { p } ) _ { \text{ins} }$. However, using the methods of [a4] and [a5], it is plausible that even in this case one can restore the theorem for $M = M ^ { \prime } \cap K _ { \operatorname { tot } S }$.

By Hilbert's Nullstellensatz (cf. also Hilbert theorem), $K _ { s }$ is PAC over $O _ { K }$. Hence, [a8], Thms.1.4; 1.5, generalize the density theorem and the local-global principle above. Probability theory supplies an abundance of other algebraic extensions of $K$ which are PAC over $O _ { K }$. The measure space in question is the Cartesian product $G ( K ) ^ { e }$ of $e$ copies of the absolute Galois group of $K$ equipped with the Haar measure. For each $\overline { \sigma } = ( \sigma _ { 1 } , \ldots , \sigma _ { e } ) \in G ( K ) ^ { e }$, let $K _ { s } ( \overline { \sigma } )$ be the fixed field of $\sigma _ { 1 } , \ldots , \sigma _ { e }$ in $K _ { s }$. By [a7], Prop. 3.1, $K _ { s } ( \overline { \sigma } )$ is PAC over $O _ { K }$ for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. Together with the preceding paragraph, this yields the following result (the Jarden–Razon theorem): For every positive integer $e$ and for almost all $\overline { \sigma } \in G ( K ) ^ { e }$, the field $M = ( K _ { s } ( \overline { \sigma } ) \cap K _ { \text{tot } S} ) _ { \text{in s} }$ satisfies the conclusions of the local-global principle and the density theorem.

The local-global principle for rings implies a local-global principle for fields. An algebraic extension $M$ of $K$ is said to be P$S$C (pseudo $S$-adically closed) if each absolutely irreducible variety $V$ over $M$ which has a simple $M _ { \operatorname{p} }$-rational point for each $\operatorname{p} \in S _ { M}$, has an $M$-rational point. In particular, by the local-global principle and the Jarden–Razon theorem, the fields $K _ { \text{tot }S }$ and $( K _ { s } ( \overline { \sigma } ) \cap K _ { totS } ) _ { ins }$ are P$S$C for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. The main result of [a5] supplies P$S$C extensions of $K$ which are even smaller than the fields $K _ { s } ( \overline { \sigma } ) \cap K _ { \operatorname{tot}S }$ (the Geyer–Jarden theorem): For every positive integer $e$ and for almost all $\overline { \sigma } \in G ( K ) ^ { e }$, the field $K _ { s } [ \overline { \sigma } ] \cap K _ { \text{tot }S }$ is P$S$C.

Here, $K _ { S } [ \overline { \sigma } ]$ is the maximal Galois extension of $K$ that is contained in $K _ { s } ( \overline { \sigma } )$. It is not known (1998) whether $O _ { K _ { s } [ \bar{\sigma} ] } $ satisfies the local-global principle. (So, the Geyer–Jarden theorem is not a consequence of the Jarden–Razon theorem.) Since a separable algebraic extension of a P$S$C field is P$S$C [a9], Lemma 7.2, the Geyer–Jarden theorem implies that $K _ { s } ( \overline { \sigma } ) \cap K _ { \operatorname{tot}S }$ is P$S$C for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. Likewise, it reproves that $K _ { \text{tot }S }$ is P$S$C.

A field $M$ which is P$S$C is also ample (i.e. if $V$ is an absolutely irreducible variety over $M$ and $V _ { \text { simp } } ( M ) \neq \emptyset$, then $V ( M )$ is Zariski-dense in $V$). Ample fields, in particular P$S$C fields, have the nice property that the inverse problem of Galois theory over $M ( t )$ has a positive solution (cf. also Galois theory, inverse problem of). That is, for every finite group $G$ there exists a Galois extension $F$ of $M ( t )$ such that $\operatorname{Gal}( F / M ( t ) ) \cong G$. Indeed, every finite split embedding problem over $M ( t )$ is solvable [a11], Main Thm. A, [a6], Thm. 2.

Another interesting consequence of the local-global principle describes the absolute Galois group of $K _ { \text{tot }S }$: It is due to F. Pop [a11], Thm. 3, and may be considered as a local-global principle for the absolute Galois group of $K _ { \text{tot }S }$ (Pop's theorem): The absolute Galois group of $K _ { \text{tot }S }$ is the free pro-finite product

\begin{equation*} \prod _ { \text{p} ^ { \prime } \in S ^ { \prime } } G ( K _ { \text{p} ^ { \prime } } ), \end{equation*}

where $S ^ { \prime }$ is the set of all extensions to $K _ { \text{tot }S }$ of all $\text{p} \in S$. This means that if $G$ is a finite group, then each continuous mapping $\alpha _ { 0 } : \cup _ { \text { p } ^ { \prime } \in S ^ { \prime } } G ( K _ { \text { p } ^ { \prime } } ) \rightarrow G$ whose restriction to each $G ( K _ { \operatorname{p} ^ { \prime } } )$ is a homomorphism, can be uniquely extended to a homomorphism $\alpha : G ( K _ { \operatorname { tot } S } ) \rightarrow G$.

As a consequence of the local-global principle, Yu.L. Ershov [a2], Thm. 3, has proved that the elementary theory of is decidable. If $S$ does not contain $\infty$, this implies, by [a1], p. 86; Corol. 10, that the elementary theory of ${\bf Z} _ { \text{tot} S }$ is decidable. In particular, Hilbert's tenth problem has an affirmative solution over ${\bf Z} _ { \text{tot} S }$. If however, $S = \{ \infty \}$, then the elementary theory of is decidable [a3] but the elementary theory of ${\bf Z} _ { \text{tot} S }$ is undecidable [a12].

References

[a1] L. Darnière, "Étude modèle-théorique d'anneaus satisfaisant un principe de Hasse non singulier" PhD Thesis (1998)
[a2] Yu.L. Ershov, "Nice local-global fields I" Algebra and Logic , 35 (1996) pp. 229–235
[a3] M.D. Fried, D. Haran, H. Völklein, "Real hilbertianity and the field of totally real numbers" Contemp. Math. , 74 (1994) pp. 1–34
[a4] B. Green, F. Pop, P. Roquette, "On Rumely's local-global principle" Jahresber. Deutsch. Math. Ver. , 97 (1995) pp. 43–74
[a5] W.-D. Geyer, M. Jarden, "PSC Galois extensions of Hilbertian fields" Manuscript Tel Aviv (1998)
[a6] D. Haran, M. Jarden, "Regular split embedding problems over function fields of one variable over ample fields" J. Algebra , 208 (1998) pp. 147–164
[a7] M. Jarden, A. Razon, "Pseudo algebraically closed fields over rings" Israel J. Math. , 86 (1994) pp. 25–59
[a8] M. Jarden, A. Razon, "Rumely's local global principle for algebraic P$\mathcal{S}$C fields over rings" Trans. Amer. Math. Soc. , 350 (1998) pp. 55–85
[a9] M. Jarden, "Algebraic realization of $p$-adically projective groups" Compositio Math. , 79 (1991) pp. 21–62
[a10] L. Moret-Bailly, "Groupes de Picard et problèmes de Skolem II" Ann. Sci. Ecole Norm. Sup. , 22 (1989) pp. 181–194
[a11] F. Pop, "Embedding problems over large fields" Ann. of Math. , 144 (1996) pp. 1–34
[a12] J. Robinson, "On the decision problem for algebraic rings" , Studies Math. Anal. Rel. Topics , Stanford Univ. Press (1962) pp. 297–304.
How to Cite This Entry:
Local-global principles for large rings of algebraic integers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local-global_principles_for_large_rings_of_algebraic_integers&oldid=39691
This article was adapted from an original article by Moshe Jarden (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article