Local-global principles for the ring of algebraic integers

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.

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