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Difference between revisions of "Local-global principles for the ring of algebraic integers"

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Consider the [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201301.png" /> of rational numbers and the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201302.png" /> of rational integers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201303.png" /> be the field of all algebraic numbers (cf. also [[Algebraic number|Algebraic number]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201304.png" /> be the ring of all algebraic integers. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201305.png" /> is the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201307.png" /> is the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201308.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l1201309.png" /> (cf. also [[Extension of a field|Extension of a field]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013010.png" /> is a [[Polynomial|polynomial]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013011.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013012.png" /> and there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013014.png" /> is a unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013015.png" />, then the greatest common divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013016.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013017.png" />. In 1934, T. Skolem [[#References|[a14]]] proved that the converse is also true (Skolem's theorem): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013018.png" /> be a primitive polynomial with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013019.png" />. Then there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013021.png" /> is a unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013022.png" />.
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Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013023.png" /> is said to be primitive if the ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013024.png" /> generated by its coefficients is the whole ring.
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E.C. Dade [[#References|[a2]]] rediscovered this theorem in 1963. D.R. Estes and R.M. Guralnick [[#References|[a5]]] reproved it in 1982 and drew some consequences about local-global principles for modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013025.png" />. In 1984, D.C. Cantor and P. Roquette [[#References|[a1]]] considered rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013026.png" /> and proved a local-global principle for the "Skolem problem with data f1…fm" (the Cantor–Roquette theorem): Suppose that for each [[Prime number|prime number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013027.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013030.png" />. Then there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013033.png" />.
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Consider the [[Field|field]] $\mathbf{Q}$ of rational numbers and the ring $\bf Z$ of rational integers. Let $\tilde {\bf Q }$ be the field of all algebraic numbers (cf. also [[Algebraic number|Algebraic number]]) and let $\widetilde{\bf Z}$ be the ring of all algebraic integers. Then $\tilde {\bf Q }$ is the algebraic closure of $\mathbf{Q}$ and $\widetilde{\bf Z}$ is the integral closure of $\bf Z$ in $\tilde {\bf Q }$ (cf. also [[Extension of a field|Extension of a field]]). If $f ( X ) = a _ { n } X ^ { n } + a _ { n - 1 } X ^ { n - 1 } + \ldots + a _ { 0 }$ is a [[Polynomial|polynomial]] in $X$ with coefficients in $\bf Z$ and there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a unit of $\widetilde{\bf Z}$, then the greatest common divisor of $a _ { 0 } , \dots , a _ { n }$ is $1$. In 1934, T. Skolem [[#References|[a14]]] proved that the converse is also true (Skolem's theorem): Let $f$ be a primitive polynomial with coefficients in $\widetilde{\bf Z}$. Then there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a unit of $\widetilde{\bf Z}$.
  
Here, writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013034.png" /> includes the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013035.png" /> is not a zero of the denominator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013036.png" />. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013037.png" /> is the algebraic closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013039.png" />-adic numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013040.png" /> is its valuation ring (cf. also [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013041.png" />-adic number]]).
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Here, $f$ is said to be primitive if the ideal of $\widetilde{\bf Z}$ generated by its coefficients is the whole ring.
  
Skolem's theorem follows from the Cantor–Roquette theorem applied to the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013042.png" /> by checking the local condition for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013043.png" />.
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E.C. Dade [[#References|[a2]]] rediscovered this theorem in 1963. D.R. Estes and R.M. Guralnick [[#References|[a5]]] reproved it in 1982 and drew some consequences about local-global principles for modules over $\widetilde{\bf Z}$. In 1984, D.C. Cantor and P. Roquette [[#References|[a1]]] considered rational functions $f _ { 1 } , \dots , f _ { m } \in {\bf Q} ( X _ { 1 } , \dots , X _ { n } )$ and proved a local-global principle for the "Skolem problem with data f1…fm" (the Cantor–Roquette theorem): Suppose that for each [[Prime number|prime number]] $p$ there exists an $\overline{x} \in \tilde { \mathbf{Q} } _ { p } ^ { n }$ such that $f _ { j } ( \overline{x} ) \in \tilde{\mathbf{Z}} _ { p } ^ { n }$, $j = 1 , \ldots , m$. Then there exists an $x \in \tilde { \mathbf{Q} } ^ { n }$ such that $f _ { j } ( \bar{x} ) \in \widetilde{\bf Z} ^ { n }$, $j = 1 , \ldots , m$.
  
One may consider the unirational variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013044.png" /> generated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013045.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013046.png" /> by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013047.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013048.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013049.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013050.png" />, then, by the Cantor–Roquette theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013051.png" />. Rumely's local-global principle [[#References|[a12]]], Thm. 1, extends this result to arbitrary varieties: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013052.png" /> be an absolutely irreducible affine variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013054.png" /> for all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013055.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013056.png" />.
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Here, writing $f _ { j } ( \overline{x} )$ includes the assumption that $\bar{x}$ is not a zero of the denominator of $f _ { j } ( \overline { X } )$. Also, $\widetilde { \mathbf{Q} }_ p$ is the algebraic closure of the field $\mathbf{Q} _ { p }$ of $p$-adic numbers and $\widetilde{\mathbf{Z}} _ { p }$ is its valuation ring (cf. also [[P-adic number|$p$-adic number]]).
  
R. Rumely has enhanced his local-global principle by a density theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013057.png" /> be an affine absolutely irreducible variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013058.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013059.png" /> be a finite set of prime numbers. Suppose that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013061.png" /> is a non-empty open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013062.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013063.png" />-adic topology, which is stable under the action of the [[Galois group|Galois group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013064.png" />. In addition, assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013065.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013066.png" />. Then there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013067.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013068.png" />, all conjugates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013069.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013070.png" /> belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013071.png" />, and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013073.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013074.png" />-integral.
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Skolem's theorem follows from the Cantor–Roquette theorem applied to the data $( X , 1 / f ( X ) )$ by checking the local condition for each $p$.
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One may consider the unirational variety $V$ generated in $A ^ { n }$ over $\mathbf{Q}$ by the $m$-tuple $( f _ { 1 } ( \overline{X} ) , \dots , f _ { m } ( \overline{X} ) )$. If $V ( \widetilde{Z} _ { p } ) \neq \emptyset$ for each $p$, then, by the Cantor–Roquette theorem, $V ( \tilde{\mathbf{Z}} ) \neq \emptyset$. Rumely's local-global principle [[#References|[a12]]], Thm. 1, extends this result to arbitrary varieties: Let $V$ be an absolutely irreducible affine variety over $\mathbf{Q}$. If $V ( \widetilde{Z} _ { p } ) \neq \emptyset$ for all prime numbers $p$, then $V ( \tilde{\mathbf{Z}} ) \neq \emptyset$.
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R. Rumely has enhanced his local-global principle by a density theorem: Let $V$ be an affine absolutely irreducible variety over $\mathbf{Q}$ and let $S$ be a finite set of prime numbers. Suppose that for each $p \in S$, $\mathcal{U} _ { p }$ is a non-empty open subset of $V ( \tilde { \mathbf{Q} } _ { p } )$ in the $p$-adic topology, which is stable under the action of the [[Galois group|Galois group]] $\operatorname{Gal}(\tilde{\mathbf{Q}_p}/\mathbf{Q}_p)$. In addition, assume that $V ( \widetilde{Z} _ { p } ) \neq \emptyset$ for all $p \notin S$. Then there exists an $\bar{x} \in V ( \tilde{\mathbf{Q}} )$ such that for each $p \in S$, all conjugates of $\bar{x}$ over $\mathbf{Q}$ belong to $\mathcal{U} _ { p }$, and for each $p \notin S$, $\bar{x}$ is $p$-integral.
  
 
The proof of this theorem uses complex-analytical methods, especially the Fekete–Szegö theorem from capacity theory. The latter is proved in [[#References|[a13]]]. See [[#References|[a9]]] for an algebraic proof of the local-global principle using the language of schemes; see [[#References|[a7]]] for still another algebraic proof of it, written in the language of classical [[Algebraic geometry|algebraic geometry]]. Both proofs enhance the theorem in various ways, see also [[Local-global principles for large rings of algebraic integers|Local-global principles for large rings of algebraic integers]].
 
The proof of this theorem uses complex-analytical methods, especially the Fekete–Szegö theorem from capacity theory. The latter is proved in [[#References|[a13]]]. See [[#References|[a9]]] for an algebraic proof of the local-global principle using the language of schemes; see [[#References|[a7]]] for still another algebraic proof of it, written in the language of classical [[Algebraic geometry|algebraic geometry]]. Both proofs enhance the theorem in various ways, see also [[Local-global principles for large rings of algebraic integers|Local-global principles for large rings of algebraic integers]].
  
As a matter of fact, all these theorems can be proved for an arbitrary number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013075.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013076.png" />. One has to replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013077.png" /> by the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013078.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013079.png" /> and the prime numbers by the non-zero prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013080.png" />. This is important for the positive solution of Hilbert's tenth problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013081.png" /> [[#References|[a12]]], Thm. 2: There is a primitive recursive procedure to decide whether given polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013082.png" /> have a common zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013083.png" />.
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As a matter of fact, all these theorems can be proved for an arbitrary number field $K$ instead of $\mathbf{Q}$. One has to replace $\bf Z$ by the ring of integers $O _ { K }$ of $K$ and the prime numbers by the non-zero prime ideals of $O _ { K }$. This is important for the positive solution of Hilbert's tenth problem for $\widetilde{\bf Z}$ [[#References|[a12]]], Thm. 2: There is a primitive recursive procedure to decide whether given polynomials $f _ { 1 } , \dots , f _ { m } \in \tilde{\bf Z} [ X _ { 1 } , \dots , X _ { n } ]$ have a common zero in $\tilde{\mathbf{Z}} ^ { n }$.
  
To this end, recall that the original Hilbert tenth problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013084.png" /> has a negative solution [[#References|[a8]]] (cf. also [[Hilbert problems|Hilbert problems]]). Similarly, the local-global principle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013085.png" /> holds only in very few cases, such as quadratic forms.
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To this end, recall that the original Hilbert tenth problem for $\bf Z$ has a negative solution [[#References|[a8]]] (cf. also [[Hilbert problems|Hilbert problems]]). Similarly, the local-global principle over $\bf Z$ holds only in very few cases, such as quadratic forms.
  
In the language of model theory (cf. also [[Model theory of valued fields|Model theory of valued fields]]), this positive solution states that the existential theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013086.png" /> is decidable in a primitive-recursive way (cf. [[#References|[a6]]], Chap. 17, for the notion of primitive recursiveness in algebraic geometry). L. van den Dries [[#References|[a4]]] has strengthened this result (van den Dries' theorem): The elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013087.png" /> is decidable.
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In the language of model theory (cf. also [[Model theory of valued fields|Model theory of valued fields]]), this positive solution states that the existential theory of $\widetilde{\bf Z}$ is decidable in a primitive-recursive way (cf. [[#References|[a6]]], Chap. 17, for the notion of primitive recursiveness in algebraic geometry). L. van den Dries [[#References|[a4]]] has strengthened this result (van den Dries' theorem): The elementary theory of $\widetilde{\bf Z}$ is decidable.
  
Indeed, van den Dries proves that each statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013089.png" /> about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013090.png" /> in the language of rings is equivalent to a quantifier-free statement about the parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013091.png" />. The latter statement, however, must be written in a language which includes extra predicates, called radicals. They express inclusion between ideals that depend on the parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013092.png" />. A special case of the main result of [[#References|[a11]]] is an improvement of van den Dries' theorem. It says that the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013093.png" /> is even primitive recursively decidable. The decision procedure is based on the method of Galois stratification [[#References|[a6]]], Chap. 25, adopted to the language of rings with radical relations.
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Indeed, van den Dries proves that each statement $\theta$ about $\widetilde{\bf Z}$ in the language of rings is equivalent to a quantifier-free statement about the parameters of $\theta$. The latter statement, however, must be written in a language which includes extra predicates, called radicals. They express inclusion between ideals that depend on the parameters of $\theta$. A special case of the main result of [[#References|[a11]]] is an improvement of van den Dries' theorem. It says that the elementary theory of $\widetilde{\bf Z}$ is even primitive recursively decidable. The decision procedure is based on the method of Galois stratification [[#References|[a6]]], Chap. 25, adopted to the language of rings with radical relations.
  
Looking for possible generalizations of the above theorems, van den Dries and A. Macintyre [[#References|[a3]]] have axiomatized the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013095.png" />. The axioms are written in the language of rings extended by the "radical relations" mentioned above.
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Looking for possible generalizations of the above theorems, van den Dries and A. Macintyre [[#References|[a3]]] have axiomatized the elementary theory of $\widetilde{\bf Z}$. The axioms are written in the language of rings extended by the "radical relations" mentioned above.
  
A. Prestel and J. Schmid [[#References|[a10]]] take another approach to the radical relations and supply another set of axioms for the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013096.png" />. Their approach yields the following analogue to Hilbert's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013098.png" />th problem for polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l12013099.png" />, which was solved by E. Artin and O. Schreier in 1927: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130100.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130101.png" /> belongs to the radical of the ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130102.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130103.png" /> if and only if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130105.png" /> belongs to the radical of the ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130106.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130107.png" />.
+
A. Prestel and J. Schmid [[#References|[a10]]] take another approach to the radical relations and supply another set of axioms for the elementary theory of $\widetilde{\bf Z}$. Their approach yields the following analogue to Hilbert's $17$th problem for polynomials over $\mathbf{R}$, which was solved by E. Artin and O. Schreier in 1927: Let $f , g _ { 1 } , \dots , g _ { m } \in \mathbf{Z} [ X _ { 1 } , \dots , X _ { n } ]$. Then $f$ belongs to the radical of the ideal generated by $g _ { 1 } , \ldots , g _ { m }$ in ${\bf Z} [ X _ { 1 } , \dots , X _ { n } ]$ if and only if for all $a \in \widetilde{\mathbf{Z}} ^ { n}$, $f ( a )$ belongs to the radical of the ideal generated by $g _ { 1 } ( a ) , \ldots , g _ { m } ( a )$ in $\widetilde{\bf Z}$.
  
Needless to say that the proofs of these theorems, as well as the axiomatizations of the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120130/l120130108.png" />, depend on Rumely's local-global principle.
+
Needless to say that the proofs of these theorems, as well as the axiomatizations of the elementary theory of $\widetilde{\bf Z}$, depend on Rumely's local-global principle.
  
 
The results mentioned above have been strongly generalized in various directions; see also [[Local-global principles for large rings of algebraic integers|Local-global principles for large rings of algebraic integers]].
 
The results mentioned above have been strongly generalized in various directions; see also [[Local-global principles for large rings of algebraic integers|Local-global principles for large rings of algebraic integers]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.C. Cantor, P. Roquette, "On diophantine equations over the ring of all algebraic integers" ''J. Number Th.'' , '''18''' (1984) pp. 1–26 {{MR|0734433}} {{ZBL|0538.12014}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.C. Dade, "Algebraic integral representations by arbitrary forms" ''Mathematika'' , '''10''' (1963) pp. 96–100 (Correction: 11 (1964), 89–90) {{MR|0166163}} {{MR|0166164}} {{ZBL|0121.28502}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. van den Dries, A. Macintyre, "The logic of Rumely's local-global principle" ''J. Reine Angew. Math.'' , '''407''' (1990) pp. 33–56 {{MR|1048527}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L. van den Dries, "Elimination theory for the ring of algebraic integers" ''J. Reine Angew. Math.'' , '''388''' (1988) pp. 189–205 {{MR|}} {{ZBL|0659.12021}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> D.R. Estes, R.M. Guralnick, "Module equivalence: local to global when primitive polynomials represent units" ''J. Algebra'' , '''77''' (1982) pp. 138–157 {{MR|0665169}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M.D. Fried, M. Jarden, "Field arithmetic" , ''Ergebn. Math. III'' , '''11''' , Springer (1986) {{MR|0868860}} {{ZBL|0625.12001}} </TD></TR><TR><TD valign="top">[a7]</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 {{MR|1341772}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> Y. Matijasevich, "Enumerable sets are diophantine" ''Soviet Math. Dokl.'' , '''11''' (1970) pp. 354–357 (In Russian)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> L. Moret-Bailly, "Groupes de Picard et problèmes de Skolem I" ''Ann. Sci. Ecole Norm. Sup.'' , '''22''' (1989) pp. 161–179 {{MR|}} {{ZBL|0704.14014}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> A. Prestel, J. Schmid, "Existentially closed domains with radical relations" ''J. Reine Angew. Math.'' , '''407''' (1990) pp. 178–201 {{MR|1048534}} {{ZBL|0691.12013}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Razon, "Primitive recursive decidability for large rings of algebraic integers" ''PhD Thesis Tel Aviv'' (1996)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> R. Rumely, "Arithmetic over the ring of all algebraic integers" ''J. Reine Angew. Math.'' , '''368''' (1986) pp. 127–133 {{MR|0850618}} {{ZBL|0581.14014}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> R. Rumely, "Capacity theory on algebraic curves" , ''Lecture Notes Math.'' , '''1378''' , Springer (1989) {{MR|1009368}} {{ZBL|0679.14012}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> Th. Skolem, "Lösung gewisser Gleichungen in ganzen algebraischen Zahlen, insbesondere in Einheiten" ''Skr. Norske Videnskaps-Akademi Oslo I. Mat. Naturv. Kl.'' , '''10''' (1934) {{MR|}} {{ZBL|0011.19701}} </TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> D.C. Cantor, P. Roquette, "On diophantine equations over the ring of all algebraic integers" ''J. Number Th.'' , '''18''' (1984) pp. 1–26 {{MR|0734433}} {{ZBL|0538.12014}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> E.C. Dade, "Algebraic integral representations by arbitrary forms" ''Mathematika'' , '''10''' (1963) pp. 96–100 (Correction: 11 (1964), 89–90) {{MR|0166163}} {{MR|0166164}} {{ZBL|0121.28502}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> L. van den Dries, A. Macintyre, "The logic of Rumely's local-global principle" ''J. Reine Angew. Math.'' , '''407''' (1990) pp. 33–56 {{MR|1048527}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> L. van den Dries, "Elimination theory for the ring of algebraic integers" ''J. Reine Angew. Math.'' , '''388''' (1988) pp. 189–205 {{MR|}} {{ZBL|0659.12021}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> D.R. Estes, R.M. Guralnick, "Module equivalence: local to global when primitive polynomials represent units" ''J. Algebra'' , '''77''' (1982) pp. 138–157 {{MR|0665169}} {{ZBL|}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> M.D. Fried, M. Jarden, "Field arithmetic" , ''Ergebn. Math. III'' , '''11''' , Springer (1986) {{MR|0868860}} {{ZBL|0625.12001}} </td></tr><tr><td valign="top">[a7]</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 {{MR|1341772}} {{ZBL|}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> Y. Matijasevich, "Enumerable sets are diophantine" ''Soviet Math. Dokl.'' , '''11''' (1970) pp. 354–357 (In Russian)</td></tr><tr><td valign="top">[a9]</td> <td valign="top"> L. Moret-Bailly, "Groupes de Picard et problèmes de Skolem I" ''Ann. Sci. Ecole Norm. Sup.'' , '''22''' (1989) pp. 161–179 {{MR|}} {{ZBL|0704.14014}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> A. Prestel, J. Schmid, "Existentially closed domains with radical relations" ''J. Reine Angew. Math.'' , '''407''' (1990) pp. 178–201 {{MR|1048534}} {{ZBL|0691.12013}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> A. Razon, "Primitive recursive decidability for large rings of algebraic integers" ''PhD Thesis Tel Aviv'' (1996)</td></tr><tr><td valign="top">[a12]</td> <td valign="top"> R. Rumely, "Arithmetic over the ring of all algebraic integers" ''J. Reine Angew. Math.'' , '''368''' (1986) pp. 127–133 {{MR|0850618}} {{ZBL|0581.14014}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> R. Rumely, "Capacity theory on algebraic curves" , ''Lecture Notes Math.'' , '''1378''' , Springer (1989) {{MR|1009368}} {{ZBL|0679.14012}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> Th. Skolem, "Lösung gewisser Gleichungen in ganzen algebraischen Zahlen, insbesondere in Einheiten" ''Skr. Norske Videnskaps-Akademi Oslo I. Mat. Naturv. Kl.'' , '''10''' (1934) {{MR|}} {{ZBL|0011.19701}} </td></tr></table>

Latest revision as of 16:57, 1 July 2020

Consider the field $\mathbf{Q}$ of rational numbers and the ring $\bf Z$ of rational integers. Let $\tilde {\bf Q }$ be the field of all algebraic numbers (cf. also Algebraic number) and let $\widetilde{\bf Z}$ be the ring of all algebraic integers. Then $\tilde {\bf Q }$ is the algebraic closure of $\mathbf{Q}$ and $\widetilde{\bf Z}$ is the integral closure of $\bf Z$ in $\tilde {\bf Q }$ (cf. also Extension of a field). If $f ( X ) = a _ { n } X ^ { n } + a _ { n - 1 } X ^ { n - 1 } + \ldots + a _ { 0 }$ is a polynomial in $X$ with coefficients in $\bf Z$ and there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a unit of $\widetilde{\bf Z}$, then the greatest common divisor of $a _ { 0 } , \dots , a _ { n }$ is $1$. In 1934, T. Skolem [a14] proved that the converse is also true (Skolem's theorem): Let $f$ be a primitive polynomial with coefficients in $\widetilde{\bf Z}$. Then there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a unit of $\widetilde{\bf Z}$.

Here, $f$ is said to be primitive if the ideal of $\widetilde{\bf Z}$ generated by its coefficients is the whole ring.

E.C. Dade [a2] rediscovered this theorem in 1963. D.R. Estes and R.M. Guralnick [a5] reproved it in 1982 and drew some consequences about local-global principles for modules over $\widetilde{\bf Z}$. In 1984, D.C. Cantor and P. Roquette [a1] considered rational functions $f _ { 1 } , \dots , f _ { m } \in {\bf Q} ( X _ { 1 } , \dots , X _ { n } )$ and proved a local-global principle for the "Skolem problem with data f1…fm" (the Cantor–Roquette theorem): Suppose that for each prime number $p$ there exists an $\overline{x} \in \tilde { \mathbf{Q} } _ { p } ^ { n }$ such that $f _ { j } ( \overline{x} ) \in \tilde{\mathbf{Z}} _ { p } ^ { n }$, $j = 1 , \ldots , m$. Then there exists an $x \in \tilde { \mathbf{Q} } ^ { n }$ such that $f _ { j } ( \bar{x} ) \in \widetilde{\bf Z} ^ { n }$, $j = 1 , \ldots , m$.

Here, writing $f _ { j } ( \overline{x} )$ includes the assumption that $\bar{x}$ is not a zero of the denominator of $f _ { j } ( \overline { X } )$. Also, $\widetilde { \mathbf{Q} }_ p$ is the algebraic closure of the field $\mathbf{Q} _ { p }$ of $p$-adic numbers and $\widetilde{\mathbf{Z}} _ { p }$ is its valuation ring (cf. also $p$-adic number).

Skolem's theorem follows from the Cantor–Roquette theorem applied to the data $( X , 1 / f ( X ) )$ by checking the local condition for each $p$.

One may consider the unirational variety $V$ generated in $A ^ { n }$ over $\mathbf{Q}$ by the $m$-tuple $( f _ { 1 } ( \overline{X} ) , \dots , f _ { m } ( \overline{X} ) )$. If $V ( \widetilde{Z} _ { p } ) \neq \emptyset$ for each $p$, then, by the Cantor–Roquette theorem, $V ( \tilde{\mathbf{Z}} ) \neq \emptyset$. Rumely's local-global principle [a12], Thm. 1, extends this result to arbitrary varieties: Let $V$ be an absolutely irreducible affine variety over $\mathbf{Q}$. If $V ( \widetilde{Z} _ { p } ) \neq \emptyset$ for all prime numbers $p$, then $V ( \tilde{\mathbf{Z}} ) \neq \emptyset$.

R. Rumely has enhanced his local-global principle by a density theorem: Let $V$ be an affine absolutely irreducible variety over $\mathbf{Q}$ and let $S$ be a finite set of prime numbers. Suppose that for each $p \in S$, $\mathcal{U} _ { p }$ is a non-empty open subset of $V ( \tilde { \mathbf{Q} } _ { p } )$ in the $p$-adic topology, which is stable under the action of the Galois group $\operatorname{Gal}(\tilde{\mathbf{Q}_p}/\mathbf{Q}_p)$. In addition, assume that $V ( \widetilde{Z} _ { p } ) \neq \emptyset$ for all $p \notin S$. Then there exists an $\bar{x} \in V ( \tilde{\mathbf{Q}} )$ such that for each $p \in S$, all conjugates of $\bar{x}$ over $\mathbf{Q}$ belong to $\mathcal{U} _ { p }$, and for each $p \notin S$, $\bar{x}$ is $p$-integral.

The proof of this theorem uses complex-analytical methods, especially the Fekete–Szegö theorem from capacity theory. The latter is proved in [a13]. See [a9] for an algebraic proof of the local-global principle using the language of schemes; see [a7] for still another algebraic proof of it, written in the language of classical algebraic geometry. Both proofs enhance the theorem in various ways, see also Local-global principles for large rings of algebraic integers.

As a matter of fact, all these theorems can be proved for an arbitrary number field $K$ instead of $\mathbf{Q}$. One has to replace $\bf Z$ by the ring of integers $O _ { K }$ of $K$ and the prime numbers by the non-zero prime ideals of $O _ { K }$. This is important for the positive solution of Hilbert's tenth problem for $\widetilde{\bf Z}$ [a12], Thm. 2: There is a primitive recursive procedure to decide whether given polynomials $f _ { 1 } , \dots , f _ { m } \in \tilde{\bf Z} [ X _ { 1 } , \dots , X _ { n } ]$ have a common zero in $\tilde{\mathbf{Z}} ^ { n }$.

To this end, recall that the original Hilbert tenth problem for $\bf Z$ has a negative solution [a8] (cf. also Hilbert problems). Similarly, the local-global principle over $\bf Z$ holds only in very few cases, such as quadratic forms.

In the language of model theory (cf. also Model theory of valued fields), this positive solution states that the existential theory of $\widetilde{\bf Z}$ is decidable in a primitive-recursive way (cf. [a6], Chap. 17, for the notion of primitive recursiveness in algebraic geometry). L. van den Dries [a4] has strengthened this result (van den Dries' theorem): The elementary theory of $\widetilde{\bf Z}$ is decidable.

Indeed, van den Dries proves that each statement $\theta$ about $\widetilde{\bf Z}$ in the language of rings is equivalent to a quantifier-free statement about the parameters of $\theta$. The latter statement, however, must be written in a language which includes extra predicates, called radicals. They express inclusion between ideals that depend on the parameters of $\theta$. A special case of the main result of [a11] is an improvement of van den Dries' theorem. It says that the elementary theory of $\widetilde{\bf Z}$ is even primitive recursively decidable. The decision procedure is based on the method of Galois stratification [a6], Chap. 25, adopted to the language of rings with radical relations.

Looking for possible generalizations of the above theorems, van den Dries and A. Macintyre [a3] have axiomatized the elementary theory of $\widetilde{\bf Z}$. The axioms are written in the language of rings extended by the "radical relations" mentioned above.

A. Prestel and J. Schmid [a10] take another approach to the radical relations and supply another set of axioms for the elementary theory of $\widetilde{\bf Z}$. Their approach yields the following analogue to Hilbert's $17$th problem for polynomials over $\mathbf{R}$, which was solved by E. Artin and O. Schreier in 1927: Let $f , g _ { 1 } , \dots , g _ { m } \in \mathbf{Z} [ X _ { 1 } , \dots , X _ { n } ]$. Then $f$ belongs to the radical of the ideal generated by $g _ { 1 } , \ldots , g _ { m }$ in ${\bf Z} [ X _ { 1 } , \dots , X _ { n } ]$ if and only if for all $a \in \widetilde{\mathbf{Z}} ^ { n}$, $f ( a )$ belongs to the radical of the ideal generated by $g _ { 1 } ( a ) , \ldots , g _ { m } ( a )$ in $\widetilde{\bf Z}$.

Needless to say that the proofs of these theorems, as well as the axiomatizations of the elementary theory of $\widetilde{\bf Z}$, depend on Rumely's local-global principle.

The results mentioned above have been strongly generalized in various directions; see also Local-global principles for large rings of algebraic integers.

References

[a1] D.C. Cantor, P. Roquette, "On diophantine equations over the ring of all algebraic integers" J. Number Th. , 18 (1984) pp. 1–26 MR0734433 Zbl 0538.12014
[a2] E.C. Dade, "Algebraic integral representations by arbitrary forms" Mathematika , 10 (1963) pp. 96–100 (Correction: 11 (1964), 89–90) MR0166163 MR0166164 Zbl 0121.28502
[a3] L. van den Dries, A. Macintyre, "The logic of Rumely's local-global principle" J. Reine Angew. Math. , 407 (1990) pp. 33–56 MR1048527
[a4] L. van den Dries, "Elimination theory for the ring of algebraic integers" J. Reine Angew. Math. , 388 (1988) pp. 189–205 Zbl 0659.12021
[a5] D.R. Estes, R.M. Guralnick, "Module equivalence: local to global when primitive polynomials represent units" J. Algebra , 77 (1982) pp. 138–157 MR0665169
[a6] M.D. Fried, M. Jarden, "Field arithmetic" , Ergebn. Math. III , 11 , Springer (1986) MR0868860 Zbl 0625.12001
[a7] B. Green, F. Pop, P. Roquette, "On Rumely's local-global principle" Jahresber. Deutsch. Math. Ver. , 97 (1995) pp. 43–74 MR1341772
[a8] Y. Matijasevich, "Enumerable sets are diophantine" Soviet Math. Dokl. , 11 (1970) pp. 354–357 (In Russian)
[a9] L. Moret-Bailly, "Groupes de Picard et problèmes de Skolem I" Ann. Sci. Ecole Norm. Sup. , 22 (1989) pp. 161–179 Zbl 0704.14014
[a10] A. Prestel, J. Schmid, "Existentially closed domains with radical relations" J. Reine Angew. Math. , 407 (1990) pp. 178–201 MR1048534 Zbl 0691.12013
[a11] A. Razon, "Primitive recursive decidability for large rings of algebraic integers" PhD Thesis Tel Aviv (1996)
[a12] R. Rumely, "Arithmetic over the ring of all algebraic integers" J. Reine Angew. Math. , 368 (1986) pp. 127–133 MR0850618 Zbl 0581.14014
[a13] R. Rumely, "Capacity theory on algebraic curves" , Lecture Notes Math. , 1378 , Springer (1989) MR1009368 Zbl 0679.14012
[a14] Th. Skolem, "Lösung gewisser Gleichungen in ganzen algebraischen Zahlen, insbesondere in Einheiten" Skr. Norske Videnskaps-Akademi Oslo I. Mat. Naturv. Kl. , 10 (1934) Zbl 0011.19701
How to Cite This Entry:
Local-global principles for the ring of algebraic integers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local-global_principles_for_the_ring_of_algebraic_integers&oldid=23889
This article was adapted from an original article by Moshe Jarden (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article