In previous work, we used an $infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prov
e the analogous result for the symmetric monoidal $infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $infty$-categorical tools suitable for working with compactly generated symmetric monoidal $infty$-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the $E_{n,p}$-local $infty$-categories.
We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse
Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that $text{Aut}(text{PSL}(2,q^2))$ for $q$ an odd prime power, cannot be realized as an unramified Galois group over $K,$ but its maximal solvable quotient can. To prove this result, we determine the ring structure of the etale cohomology ring $H^*(text{Spec }mathcal{O}_K;mathbb{Z}/ 2mathbb{Z})$ where $mathcal{O}_K$ is the ring of integers of an arbitrary totally imaginary number field $K.$
