Abstract

Abstract.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. THEOREMLet G be a group of order ln, where l is a prime number. Let K0be either a finite field with |K0| >l4n+4or a pseudo finite field. Suppose that l≠ char(K0)and that K0does not contain the root of unity ζlof order l. Let K=K0(t),with t transcendental over K0.Then K has a Galois extension L with the following properties:(a)(L/K) ≅G; (b)L/K0is a regular extension; (c) genus(L) <; (d)K0[t]has exactly n prime ideals which ramify in L; the degree of each of them is[K0:K0]; (e) (t)∞totally decomposes in L; (f)L=K(x),withanddeg(ai(t)) < deg(a1(t))for i= 1,…,ln.

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