Local-global principles for the ring of algebraic integers
Consider the field of rational numbers and the ring
of rational integers. Let
be the field of all algebraic numbers (cf. also Algebraic number) and let
be the ring of all algebraic integers. Then
is the algebraic closure of
and
is the integral closure of
in
(cf. also Extension of a field). If
is a polynomial in
with coefficients in
and there exists an
such that
is a unit of
, then the greatest common divisor of
is
. In 1934, T. Skolem [a14] proved that the converse is also true (Skolem's theorem): Let
be a primitive polynomial with coefficients in
. Then there exists an
such that
is a unit of
.
Here, is said to be primitive if the ideal of
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 . In 1984, D.C. Cantor and P. Roquette [a1] considered rational functions
and proved a local-global principle for the "Skolem problem with data f1…fm" (the Cantor–Roquette theorem): Suppose that for each prime number
there exists an
such that
,
. Then there exists an
such that
,
.
Here, writing includes the assumption that
is not a zero of the denominator of
. Also,
is the algebraic closure of the field
of
-adic numbers and
is its valuation ring (cf. also
-adic number).
Skolem's theorem follows from the Cantor–Roquette theorem applied to the data by checking the local condition for each
.
One may consider the unirational variety generated in
over
by the
-tuple
. If
for each
, then, by the Cantor–Roquette theorem,
. Rumely's local-global principle [a12], Thm. 1, extends this result to arbitrary varieties: Let
be an absolutely irreducible affine variety over
. If
for all prime numbers
, then
.
R. Rumely has enhanced his local-global principle by a density theorem: Let be an affine absolutely irreducible variety over
and let
be a finite set of prime numbers. Suppose that for each
,
is a non-empty open subset of
in the
-adic topology, which is stable under the action of the Galois group
. In addition, assume that
for all
. Then there exists an
such that for each
, all conjugates of
over
belong to
, and for each
,
is
-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 instead of
. One has to replace
by the ring of integers
of
and the prime numbers by the non-zero prime ideals of
. This is important for the positive solution of Hilbert's tenth problem for
[a12], Thm. 2: There is a primitive recursive procedure to decide whether given polynomials
have a common zero in
.
To this end, recall that the original Hilbert tenth problem for has a negative solution [a8] (cf. also Hilbert problems). Similarly, the local-global principle over
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 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
is decidable.
Indeed, van den Dries proves that each statement about
in the language of rings is equivalent to a quantifier-free statement about the parameters of
. 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
. A special case of the main result of [a11] is an improvement of van den Dries' theorem. It says that the elementary theory of
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 . 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 . Their approach yields the following analogue to Hilbert's
th problem for polynomials over
, which was solved by E. Artin and O. Schreier in 1927: Let
. Then
belongs to the radical of the ideal generated by
in
if and only if for all
,
belongs to the radical of the ideal generated by
in
.
Needless to say that the proofs of these theorems, as well as the axiomatizations of the elementary theory of , 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 |
[a2] | E.C. Dade, "Algebraic integral representations by arbitrary forms" Mathematika , 10 (1963) pp. 96–100 (Correction: 11 (1964), 89–90) |
[a3] | L. van den Dries, A. Macintyre, "The logic of Rumely's local-global principle" J. Reine Angew. Math. , 407 (1990) pp. 33–56 |
[a4] | L. van den Dries, "Elimination theory for the ring of algebraic integers" J. Reine Angew. Math. , 388 (1988) pp. 189–205 |
[a5] | D.R. Estes, R.M. Guralnick, "Module equivalence: local to global when primitive polynomials represent units" J. Algebra , 77 (1982) pp. 138–157 |
[a6] | M.D. Fried, M. Jarden, "Field arithmetic" , Ergebn. Math. III , 11 , Springer (1986) |
[a7] | B. Green, F. Pop, P. Roquette, "On Rumely's local-global principle" Jahresber. Deutsch. Math. Ver. , 97 (1995) pp. 43–74 |
[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 |
[a10] | A. Prestel, J. Schmid, "Existentially closed domains with radical relations" J. Reine Angew. Math. , 407 (1990) pp. 178–201 |
[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 |
[a13] | R. Rumely, "Capacity theory on algebraic curves" , Lecture Notes Math. , 1378 , Springer (1989) |
[a14] | Th. Skolem, "Lösung gewisser Gleichungen in ganzen algebraischen Zahlen, insbesondere in Einheiten" Skr. Norske Videnskaps-Akademi Oslo I. Mat. Naturv. Kl. , 10 (1934) |
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=18276