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.