View source for Local-global principles for the ring of algebraic integers

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

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 $f _ { j } ( \bar{x} ) \in \widetilde{\bf Z} ^ { n }$, $j = 1 , \ldots , m$.

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]]).

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 [[#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 $V ( \tilde{\mathbf{Z}} ) \neq \emptyset$.

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]].

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 $\bf Z$ 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 $\widetilde{\bf Z}$ 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 $\theta$ 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 $\widetilde{\bf Z}$. 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 $\widetilde{\bf Z}$. 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|Local-global principles for large rings of algebraic integers]].

====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>