Hilbertianity of division fields of commutative algebraic groups. (English) Zbl 1379.12004
Summary: Let \(G\) be a commutative algebraic group over a finitely generated infinite field \(K\) of characteristic \(p\). We prove that every extension of \(K\) contained in the field obtained by adjoining to \(K\) all prime-to-\(p\) torsion points of \(G\) is Hilbertian. We also determine when the field obtained by adjoining to \(K\) all torsion points of \(G\) has this property. This extends results of Moshe Jarden on abelian varieties.
MSC:
| 12E25 | Hilbertian fields; Hilbert’s irreducibility theorem |
| 20G99 | Linear algebraic groups and related topics |
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