Abstract
We show that the theory of Frobenius fields is decidable. This is conjectured in [4], [8] and [13], and we prove it by solving a group theoretic problem to which this question is reduced there. To do this we present and develop the notion of embedding covers of finite and pro-finite groups. We also answer two other problems from [8], again by solving a corresponding group theoretic problem: A finite extension of a Frobenius field need not be Frobenius and there are PAC fields which are not Frobenius fields.
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References
J. Ax,Solving diophantine problems modulo every prime, Ann. of Math.85 (1967), 161–183.
J. Ax,The elementary theory of finite fields, Ann. of Math.86 (1968), 239–271.
G. Cherlin, A. Macintyre and L. van den Dries,Decidability and undecidability theorems for PAC fields, Bull. Amer. Math. Soc.4 (1981), 101–104.
G. Cherlin, A. Macintyre and L. van den Dries,The elementary theory of p.a.c. fields, a preprint.
J. Cossey, O. H. Kegel and L. G. KovácsMaximal Frattini extensions, Arch. Math.35 (1980), 210–217.
Yu. L. Ershov,Regularly closed fields, Doklady251, No. 4 (1980), 783–785.
Yu. Ershov and M. Fried,Frattini covers and projective groups without the extension property, Math. Ann.253 (1980), 233–239.
M. Fried, D. Haran and M. Jarden,Galois stratification over Frobenius fields, Advances in Math., to appear.
M. Fried and G. Sacerdote,Solving diophantine problems over all residue class fields and all finite fields, Ann. of Math.104 (1976), 203–233.
K. Gruenberg,Projective profinite groups, J. London Math. Soc.42 (1967), 155–165.
M. Jarden,Elementary statements over large algebraic fields, Trans. Amer. Math. Soc.164 (1972), 67–91.
M. Jarden,The elementary theory of ω-free Ax fields, Invent. Math.38 (1976), 187–206.
M. Jarden,Normal Frobenius fields, a preprint.
A. Lubotzky and L. van den Dries,Subgroups of free profinite groups and large subfields of Q, Israel J. Math.39 (1981), 25–45.
P. Ribes,Introduction of profinite groups and Galois cohomology, Queen Papers in Pure and Applied Mathematics24, Queen's University, Kingston, Ontario, 1970.
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Portions of this work will be incorporated in the doctoral dissertation of the first author done in Tel Aviv University under the supervision of Prof. Moshe Jarden.
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Haran, D., Lubotzky, A. Embedding covers and the theory of frobenius fields. Israel J. Math. 41, 181–202 (1982). https://doi.org/10.1007/BF02771720
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DOI: https://doi.org/10.1007/BF02771720