Florian Pop

Embedding problems over large fields

In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on varieties. As applications, we give a positive answer to a long standing conjecture which originates in an unpublished note of Roquette, and asserts that the absolute Galois group of a countable, hilbertian, PAC field is profinite free; see [F-J, Problem 24.41]. Secondly, we give new evidence for the conjecture of Shafarevich which asserts that the absolute Galois group of Qab is profinite free. Finally, one of the most interesting applications of the theory we develop here is the insight in the Galois structure of the totally EG-adic numbers.

Valuation theory and its applications

A note on tame fields by K. Aghigh and S. K. Khanduja Some remarks about asymptotic couples by M. Aschenbrenner Prime segments for cones and rings by H. H. Brungs, H. Marubayashi, and E. Osmanagic Irreducibility criterion: A geometric point of view by V. Cossart and G. Moreno-Socias On the decidability of the existential theory of

Elementary equivalence versus isomorphism

{\mathbb F_p}[[t]]

Alterations and Birational Anabelian Geometry

by J. Denef and H. Schoutens Galois groups over nonrigid fields by W. Gao, D. B. Leep, J. Minac, and T. L. Smith Automorphisms of formal power series rings over a valuation ring by B. Green Regular curves over Prufer domains by H. Knaf Encoding valuations in absolute Galois groups by J. Koenigsmann Dynamic computations inside the algebraic closure of a valued field by F.-V. Kuhlmann, H. Lombardi, and H. Perdry Preorders, rings, lattice-ordered groups and formal power series by G. Leloup The theorem of Grunwald-Wang in the setting of valuation theory by F. Lorenz and P. Roquette Invariants of singular plane curves by R. I. Michler

Projective group structures as absolute Galois structures with block approximation

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Characterization of extremal valued fields

-rational fields by J. Ohm A generalization of Hensels lemma by H. Perdry Classically projective groups and pseudo classically closed fields by F. Pop Approximate roots by P. Popescu-Pampu Quantifier elimination for the relative Frobenius by T. Scanlon Ultrametric fixed point theorems and applications by E. Schorner Valuations, deformations, and toric geometry by B. Teissier.

The absolute Galois group of the field of totally

How does the first order language of fields encode birational invariants of varieties?... This question is related to rational points on varieties and effectiveness in algebraic/arithmetic geometry.

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The basic idea of Grothendieck’s anabelian geometry is that under certain “anabelian hypotheses” the etale fundamental group of a scheme contains all the geometric and arithmetic information about the scheme in discussion, that is to say, the scheme isfunctorially encodedin its etale fundamental group. Such ideas are not completely new, the first assertion of this type being the celebrated result of Artin-Schreier from the middle of the Twenties, which asserts that if the absolute Galois group of some field is non-trivial and finite, then the field in discussion is real closed. This is nevertheless not an assertion about the structure/isomorphy type of the field in discussion, but rather about its elementary theory. It was the attempt to give ap-adic analogueof the Artin-Schreier Theorem which lead Neukirch to the question whether the isomorphy type of a number field (as a field) isgroup theoreticallyencoded in the isomorphy type of the absolute Galois group (as a profinite group) of the number field in discussion.

-adic numbers

Introduction Etale topology Group structures Completion of a cover to a cartesian square Projective group structures Special covers Unirationally closed fields Valued fields The space of valuation of a field Locally uniform

First-order definitions in function fields over anti-Mordellic fields

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