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$p$-groups have unbounded realization multiplicity
In this paper we interpret the solutions to a particular Galois embedding problem over an extension $K/F$ satisfying $\operatorname {Gal}(K/F) \simeq \mathbb {Z}/p^n\mathbb {Z}$ in terms of certain Galois submodules within the parameterizing space of elementary $p$-abelian extensions of $K$; here $p$ is a prime. Combined with some basic facts …