How often can a finite group be realized as a Galois group over a field?

A recursive description of pro-p Galois groups

“…✷ Examples of fields of the type we studied in this section are provided by algebraic extensions of global fields [3,Main Theorem]. For some other examples see §1.2 of [13].…”

Parameterizing solutions to any Galois embedding problem overZ/pnZwith elementary p -abelian kernel

“…This latter quantity is called the realization multiplicity of G. Jensen has explored realization multiplicities in [17,18]. We have a generalization of the main result from [1].…”

“…So suppose we have chosen a filtration F W that satisfies (16), and let I be a basis for W as in the proof of Lemma 5.16. A submodule U ⊆ A has F W = F len U and ( 17) if and only if U = ⊕ x∈I α x for elements {α x } ⊆ A satisfying (18) Ψ ℓ−1 α x = x, for all x ∈ I ℓ e(α x ) = 0 for all x ∈ I ℓ with ℓ < µ e(α x ) = 0 for some x ∈ I µ .…”

“…To enumerate sets {α x } satisfying (18), start by choosing elements {β x } x∈I so that Ψ ℓ−1 β x = x for all x ∈ I ℓ e(β x ) = 0 for all x ∈ I ℓ with ℓ < p n .…”

“…Now suppose that {α x } and { αx } are two collections satisfying (18); we examine when ⊕ α x = ⊕ αx . Note that for each x ∈ I ℓ there exists g x ∈ A so that αx = g x + α x , and that ℓ(g x ) < ℓ.…”