The local dimension of a finite group over a number field

Let G G be a finite group and K"> K encoding="application/x-tex">K number field. We construct -extension E slash upper F"> E / F encoding="application/x-tex">E/F , with encoding="application/x-tex">F of transcendence degree alttext="2"> 2 encoding="application/x-tex">2 over that specializes to all -extensions K Subscript German p"> p encoding="application/x-tex">K_\mathfrak {p} where alttext="German encoding="application/x-tex">\mathfrak runs but finitely many primes . If furthermore has generic extension we show the so-called Hilbert–Grunwald property. These results are compared notion essential dimension its arithmetic analogue.