Hilbertian Fields and Profinite Iterated Monodromy Groups Over Transcendental Elements with Specializations of Infinite Index

Given a field L and polynomial $f(x)/L$, one can consider the indices of the profinite iterated monodromy groups formed over the ground field within the profinite iterated monodromy groups over elements transcendental over L. For any countable Hilbertian field k, we show random Galois fixed field extensions almost surely make the indices infinite over all ground field elements, with certain conditions on $f(x)$. This is a generalization of a result in arXiv:2012.03076 by Dittmann and Kadets.