## 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. THEOREM Let G be a group of order l^{n}, where l is a prime number. Let K_{0} be either a finite field with |K_{o}| > l^{4n+4} or a pseudo finite field. Suppose that l ≠ char (K_{0}) and that K_{0} does not contain the root of unity ζ_{l} of order l. Let K = K_{0}(t), with t transcendental over K_{0}. Then K has a Galois extension L with the following properties: (a) G(L/K) ≅ G; (b) L/K_{0} is a regular extension; (c) genus(L) < 1/2nl^{2n}; (d) K_{0}[t] has exactly n prime ideals which ramify in L; the degree of each of them is [K_{0}(ζ_{ln}) : K_{0}]; (e) (t)∞ totally decomposes in L; (f) L = K(x), with irr(x, K) = X_{ln} + a_{1}(t)X_{ln - 1}1 + ⋯ + a_{ln}(t), 0 < deg(a_{1}(t)) ≤ 1/2nl^{2n} and deg(a_{i}(t)) < deg(a_{1}(t)) for i = 1, . . . , l^{n}.

Original language | English |
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Pages (from-to) | 13-62 |

Number of pages | 50 |

Journal | Nagoya Mathematical Journal |

Volume | 150 |

DOIs | |

State | Published - Jun 1998 |