The elementary theory of normal Frobenius fields
Let K be a given Hilbertian field with an elimination theory, and let M be a Frobenius field that contains K and has absolute Galois group G(M). For a profinite group G, denote by Im G the set of all finite quotient groups of G. The author of the present paper generalizes previous results of \textit{M. Fried, D. Haran} and himself [Adv. Math. 51, 1-35 (1984)] as follows. Consider a class \(\Pi\) of profinite groups, each of which appears as the absolute Galois group of a Frobenius field M. This class is supposed to be equipped with a primitive recursive algorithm to determine for given \(m+n\) finite groups \(G_ 1,...,G_ m\), \(H_ 1,...,H_ n\) whether or not there exists a \(P\in \Pi\) such that \(G_ 1,...,G_ m\in Im P\) and \(H_ 1,...,H_ n\not\in Im P\). Denoting by \({\mathcal M}\) the class of all perfect Frobenius fields M that contain K and satisfy G(M)\(\in \Pi\), it is shown that there exists a primitive recursive procedure for the theory of \({\mathcal M}.\) Using this fact and previous results of \textit{O. V. Mel'nikov} [Math. USSR, Izv. 12, 1-20 (1978); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 3-25 (1978; Zbl 0382.20032)], it is shown that the theory of normal Frobenius fields that contain K is primitive recursive via Galois stratification. Finally, using a recent result of \textit{D. Haran} and \textit{A. Lubotzky} [Isr. J. Math. 41, 181-202 (1982; Zbl 0502.20013)], it follows that the theory of all perfect Frobenius fields that contain K is decidable. - The paper ends with a rich bibliography.
theory of all perfect Frobenius fields is decidable
Hilbertian field with an elimination theory
scientific article; zbMATH DE number 3849369