TY - JOUR

T1 - Strong approximation theorem for absolutely irreducible varieties over the compositum of all symmetric extensions of a global field

AU - Jarden, Moshe

AU - Razon, Aharon

N1 - Publisher Copyright:
© Glasgow Mathematical Journal Trust 2018 A.

PY - 2019/5/1

Y1 - 2019/5/1

N2 - Let K be a global field, a proper subset of the set of all primes of K, a finite subset of , and (resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K with. Let Gal(K) = Gal(Ksep/K) be the absolute Galois group of K. For each , we choose a Henselian (respectively, a real or algebraic) closure of K at in if is non-Archimedean (respectively, archimedean). Then, is the maximal Galois extension of K in Ksep in which each totally splits. For each , we choose a-Adic absolute value of and extend it in the unique possible way to. Finally, we denote the compositum of all symmetric extensions of K by Ksymm. We consider an affine absolutely integral variety V in. Suppose that for each there exists a simple-rational point of V and for each there exists such that in both cases if is non-Archimedean and <![CDATA[\mathbf{z}-\mathfrak{p}|-\mathfrak{p} if is archimedean. Then, there exists such that for all and for all τ â Gal(K), we have if is archimedean and <![CDATA[\mathbf{z}^\tau|-\mathfrak{p} if is non-Archimedean. For , we get as a corollary that the ring of integers of Ksymm is Hilbertian and Bezout.

AB - Let K be a global field, a proper subset of the set of all primes of K, a finite subset of , and (resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K with. Let Gal(K) = Gal(Ksep/K) be the absolute Galois group of K. For each , we choose a Henselian (respectively, a real or algebraic) closure of K at in if is non-Archimedean (respectively, archimedean). Then, is the maximal Galois extension of K in Ksep in which each totally splits. For each , we choose a-Adic absolute value of and extend it in the unique possible way to. Finally, we denote the compositum of all symmetric extensions of K by Ksymm. We consider an affine absolutely integral variety V in. Suppose that for each there exists a simple-rational point of V and for each there exists such that in both cases if is non-Archimedean and <![CDATA[\mathbf{z}-\mathfrak{p}|-\mathfrak{p} if is archimedean. Then, there exists such that for all and for all τ â Gal(K), we have if is archimedean and <![CDATA[\mathbf{z}^\tau|-\mathfrak{p} if is non-Archimedean. For , we get as a corollary that the ring of integers of Ksymm is Hilbertian and Bezout.

UR - http://www.scopus.com/inward/record.url?scp=85049893838&partnerID=8YFLogxK

U2 - 10.1017/S0017089518000253

DO - 10.1017/S0017089518000253

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85049893838

VL - 61

SP - 373

EP - 380

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

SN - 0017-0895

IS - 2

ER -