Difference between revisions of "Local-global principles for large rings of algebraic integers"
(Importing text file) |
m (link) |
||
Line 23: | Line 23: | ||
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" />. | + | 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" />. |
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. | 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. |
Revision as of 17:16, 7 November 2016
Let be a global field. In other words,
is either a number field, i.e. a finite extension of
, or a function field of one variable over a finite field. Denote the algebraic (respectively, separable) closure of
by
(respectively, by
; cf. also Extension of a field). A prime divisor of
is an equivalence class
of absolute values (cf. also Norm on a field). For each
, let
be a representative of
. Denote the completion of
at
by
. Then
is either
or
(
is metric), or
is a finite extension of
or a finite extension of
(
is ultra-metric).
There is a natural -topology on
whose basic
-open subsets have the form
, for
and
,
. The
-topology has compatible extensions to all sets
, where
is an affine algebraic set over
. In each case,
is locally compact.
Embed into the algebraic closure of
and let
. Then
is a real (respectively, algebraic) closure of
at
if
(respectively,
), and is a Henselization of
at
if
is ultra-metric (cf. also Henselization of a valued field). In the latter case, the valuation ring of
is denoted by
. In each case,
is uniquely determined up to a
-isomorphism.
If is a set of prime divisors of
and
is an algebraic extension of
, then
denotes the set of all extensions to
of all
.
In the sequel, let be a fixed set of prime divisors of
which does not contain all prime divisors. For each algebraic extension
of
and each
, let
. Thus, if
is metric, then
is the
-unit ball and if
is ultra-metric, then
is the valuation ring of
. Let
. If
consists of ultra-metric primes only, then
is a Dedekind domain (cf. also Dedekind ring). For example, if
and
consists of all prime numbers, then
.
Fix also a finite subset of
. Consider the field of totally
-adic numbers:
![]() |
This is the largest Galois extension of in which each
totally splits. Let
. If
,
and
is empty, then
and
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 . Consider an absolutely irreducible affine variety
over
. Suppose that
for each
. Suppose further that
for each
. Then
.
Here, is the Zariski-open subset of
consisting of all non-singular points.
The density theorem: Let and
be as in the local-global principle. Let
be a finite subset of
containing
. Suppose that
for each
. For each
, let
be a non-empty
-open subset of
. For each
, let
be a non-empty
-open subset of
. Then
contains a point which lies in
for each
.
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 which are much smaller than
. To this end, call a field extension
of
PAC over
if for every dominating separable rational mapping
of absolutely irreducible varieties of dimension
over
, there exists an
such that
. If
is a number field and
consists of ultra-metric primes only, [a8], Thm. 1.4; 1.5, imply both the density theorem and the local-global principle for
. In the function field case, [a8] must replace
by its maximal purely inseparable extension, which is denoted by
. Accordingly, the fields
in the assumption of the density theorem and the local-global principle must be replaced by
. However, using the methods of [a4] and [a5], it is plausible that even in this case one can restore the theorem for
.
By Hilbert's Nullstellensatz (cf. also Hilbert theorem), is PAC over
. 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
which are PAC over
. The measure space in question is the Cartesian product
of
copies of the absolute Galois group of
equipped with the Haar measure. For each
, let
be the fixed field of
in
. By [a7], Prop. 3.1,
is PAC over
for almost all
. Together with the preceding paragraph, this yields the following result (the Jarden–Razon theorem): For every positive integer
and for almost all
, the field
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 of
is said to be P
C (pseudo
-adically closed) if each absolutely irreducible variety
over
which has a simple
-rational point for each
, has an
-rational point. In particular, by the local-global principle and the Jarden–Razon theorem, the fields
and
are P
C for almost all
. The main result of [a5] supplies P
C extensions of
which are even smaller than the fields
(the Geyer–Jarden theorem): For every positive integer
and for almost all
, the field
is P
C.
Here, is the maximal Galois extension of
that is contained in
. It is not known (1998) whether
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
C field is P
C [a9], Lemma 7.2, the Geyer–Jarden theorem implies that
is P
C for almost all
. Likewise, it reproves that
is P
C.
A field which is P
C is also ample (i.e. if
is an absolutely irreducible variety over
and
, then
is Zariski-dense in
). Ample fields, in particular P
C fields, have the nice property that the inverse problem of Galois theory over
has a positive solution (cf. also Galois theory, inverse problem of). That is, for every finite group
there exists a Galois extension
of
such that
. Indeed, every finite split embedding problem over
is solvable [a11], Main Thm. A, [a6], Thm. 2.
Another interesting consequence of the local-global principle describes the absolute Galois group of : It is due to F. Pop [a11], Thm. 3, and may be considered as a local-global principle for the absolute Galois group of
(Pop's theorem): The absolute Galois group of
is the free pro-finite product
![]() |
where is the set of all extensions to
of all
. This means that if
is a finite group, then each continuous mapping
whose restriction to each
is a homomorphism, can be uniquely extended to a homomorphism
.
As a consequence of the local-global principle, Yu.L. Ershov [a2], Thm. 3, has proved that the elementary theory of is decidable. If
does not contain
, this implies, by [a1], p. 86; Corol. 10, that the elementary theory of
is decidable. In particular, Hilbert's tenth problem has an affirmative solution over
. If however,
, then the elementary theory of
is decidable [a3] but the elementary theory of
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![]() |
[a9] | M. Jarden, "Algebraic realization of ![]() |
[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. |
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=16794