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Moshe Jarden

Publications and source records attributed to Moshe Jarden.

6 recordsLinked to original sources
arXiv2015

On decidable algebraic fields

We prove the following propositions. Theorem 1: Let $M$ be a subfield of a fixed algebraic closure $\tilde \Q$ of $\Q$ whose existential elementary theory is decidable (resp. primitively decidable). Then, M is conjugate to a recursive (resp. primitive recursive) subfield $L \subset \tilde \Q$. Theorem 2: For each positive integer $e$ there are infinitely many $e$-tuples $\boldsymbol σ\in \Gal(\Q)^e$ such that the field $\tilde \Q( {\boldsymbol σ})$ -- the fixed field of $\boldsymbol σ$, is recursive in $\tilde\Q$ and its elementary theory is decidable. Moreover, $\tilde \Q(\boldsymbol σ)$ is PAC and $\Gal(\tilde\Q(\boldsymbol σ))$ is isomorphic to the free profinite group on $e$ generators.

math.LO↗
arXiv2014

The Sylow subgroups of the absolute Galois group Gal(Q)

We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite Z_p-quotients of F and more generally show that every split embedding problem of Z_p-groups for F is solvable. Moreover, we analyze the Z_p-action on generators of F.

math.NT↗
arXiv2012

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 $q$.

math.NT↗
arXiv2007

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 $K$.

math.NT↗