Random Galois extensions of Hilbertian fields
Let L be a Galois extension of a countable Hilbertian field K. Although L need not be Hilbertian, we prove that an abundance of large Galois subextensions of L/K are.
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arXiv subjects
Lior Bary-Soroker
Publications and source records attributed to Lior Bary-Soroker.
On the Bateman-Horm Conjecture about Polynomial Rings
Given a power $q$ of a prime number $p$ and "nice" polynomials $f_1,...,f_r\in\bbF_q[T,X]$ with $r=1$ if $p=2$, we establish an asymptotic formula for the number of pairs $(a_1,a_2)\in\bbF_q^2$ such that $f_1(T,a_1T+a_2),...,f_r(T,a_1T+a_2)$ are irreducible in $\bbF_q[T]$. In particular that number tends to infinity with $q$.
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Subgroup structure of fundamental groups in positive characteristic
Let $Π$ be the étale fundamental group of a smooth affine curve over an algebraically closed field of characteristic $p>0$. We establish a criterion for profinite freeness of closed subgroups of $Π$. Roughly speaking, if a closed subgroup of $Π$ is "captured" between two normal subgroups, then it is free, provided it contains most of the open subgroups of index $p$. In the proof we establish a strong version of "almost $ω$-freeness" of $Π$ and then apply the Haran-Shapiro induction.
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Open Problems in the Theory of Ample Fields
Fifteen years after their discovery, ample fields now stand at the center of research in contemporary Galois theory and attract more and more attention also from other areas of mathematics. This survey gives an introduction to the theory of ample fields and discusses open problems.
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Subgroups of profinite surface groups
We study the subgroup structure of the étale fundamental group $Π$ of a projective curve over an algebraically closed field of characteristic 0. We obtain an analog of the diamond theorem for $Π$. As a consequence we show that most normal subgroups of infinite index are semi-free. In particular every proper open subgroup of a normal subgroup of infinite index is semi-free.
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Irreducibility and embedding problems
We study irreducible specializations, in particular when group-preserving specializations may not exist. We obtain a criterion in terms of embedding problems. We include several applications to analogs of Schinzel's hypothesis H and to the theory of Hilbertian fields.
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Irreducible values of polynomials
Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove an arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields. A main tool in the proof is an irreducibility theorems à la Hilbert.
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On p-embedding problems in characteristic p
Let K be a valued field of characteristic p>0 with non-p-divisible value group. We show that every finite embedding problem for K whose kernel is a p-group is properly solvable.
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Permanence criteria for semi-free profinite groups
We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.
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Pseudo Algebraically Closed Extensions
This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that although there are many PAC extensions, the Galois closure of a proper PAC extension is separably closed. The dissertation also contains the following subjects. The group theoretical counterpart of pseudo algebraically closed extensions, the so-called projective pairs. Applications to seemingly unrelated subjects, e.g., an analog of Dirichlet's theorem about primes in arithmetic progression for polynomial rings in one variable over infinite fields.
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On Pseudo Algebraically Closed Extensions of Fields
The notion of `Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. It was originally motivated by Hilbert's tenth problem, and recently had new applications. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure. This vastly generalizes earlier works of Jarden-Razon, Jarden, and Jarden and the author. This also leads to a classification of all finite PAC extensions which in turn proves the `bottom conjecture' for finitely generated infinite fields. The secondary goal of this work is to unify proofs of known results about PAC extensions and to establish new basic properties of PAC extensions, e.g.\ transitiveness of PAC extensions.
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On totally ramified extensions of discrete valued fields
We give a simple characterization of the totally wild ramified valuations in a Galois extension of fields of characteristic p. This criterion involves the valuations of Artin-Schreier cosets of the F_{p^r}^\times-translation of a single element. We apply the criterion to construct some interesting examples.
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Fully Hilbertian Fields
We introduce the notion of fully Hilbertian fields, a strictly stronger notion than that of Hilbertian fields. We show that this class of fields exhibits the same good behavior as Hilbertian fields, but for fields of uncountable cardinality, is more natural than the notion of Hilbertian fields. In particular, we show it can be used to achieve stronger Galois theoretic results. Our proofs also provide a step toward the so-called Jarden-Lubotzky twinning principle.
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On the remainder in the Taylor theorem
We give a short straightforward proof for the bound of the reminder term in the Taylor theorem. The proof uses only induction and the fact that $f'\geq 0$ implies the monotonicity of $f$, so it might be an attractive proof to give to undergraduate students.
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Projective pairs of profinite groups
We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projective pair can be realized as absolute Galois groups of a PAC extension of a PAC field. Using this characterization we construct new examples of PAC extensions of relatively small fields, e.g., unbounded abelian extensions of the rational numbers.
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On PAC Extensions and Scaled Trace Forms
Any non-degenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields. In particular, prosolvable and prime-to-p extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions.
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PAC Fields over Finitely Generated Fields
We prove the following theorem for a finitely generated field $K$: Let $M$ be a Galois extension of $K$ which is not separably closed. Then $M$ is not PAC over $K$.
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