Bounded realization of l-groups over global fields: The method of Scholz and Reichardt

We use the method of Scholz and Reichardt and a transfer principle from finite fields to pseudo finite fields in order to prove the following result. THEOREM Let G be a group of order lⁿ, where l is a prime number. Let K₀ be either a finite field with |K_o| > l⁴ⁿ⁺⁴ or a pseudo finite field. Suppose that l ≠ char (K₀) and that K₀ does not contain the root of unity ζ_l of order l. Let K = K₀(t), with t transcendental over K₀. Then K has a Galois extension L with the following properties: (a) G(L/K) ≅ G; (b) L/K₀ is a regular extension; (c) genus(L) < 1/2nl²ⁿ; (d) K₀[t] has exactly n prime ideals which ramify in L; the degree of each of them is [K₀(ζ_ln) : K₀]; (e) (t)∞ totally decomposes in L; (f) L = K(x), with irr(x, K) = X_ln + a₁(t)X_{ln - 1}1 + ⋯ + a_ln(t), 0 < deg(a₁(t)) ≤ 1/2nl²ⁿ and deg(a_i(t)) < deg(a₁(t)) for i = 1, . . . , lⁿ.