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The work  deals with questions of first-order definability in algebraic function fields. In particular, it exhibits new cases in which the field of constant functions is definable, and it investigates the phenomenon of definable transcendental elements. We fix some of its proofs and make additional observations concerning definable closure in these fields.
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 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 ζl of 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), withand deg(ai(t)) < deg(a1(t)) for i = 1,…,ln.
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