Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.3561

Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic

Martin Bays·Boris Zilber
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard kernel", are determined up to isomorphism of the algebraic structure by the characteristic and transcendence degree of $F$ and, in positive characteristic, the restriction of the cover to finite fields. This extends the main result of "Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero" (B. Zilber, JLMS 2007), and our proof fills a hole in the proof given there.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Martin Bays, Boris Zilber. 2011-01-04. Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic. https://doi.org/10.1112/blms%2Fbdq131

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Borel sets in ideal topologies

We study the Borel and analytic subsets of the spaces ${}^κκ$ and ${}^κ2$ endowed with ideal topologies, where $κ$ is a regular uncountable cardinal, thereby addressing some open problems of the literature. We provide a systematic analysis of the Borel hierarchy for an arbitrary ideal topology. In particular, we formulate a sufficient condition ensuring that the hierarchy does not collapse, demonstrate that every Borel set in such a topology is analytic, and establish the existence of a set that is not Borel. Our main result shows that, when the underlying ideal contains an unbounded subset, the collection of analytic sets coincides with the full power set of the ambient space. Finally, we prove that the Approximation Lemma holds in the setting of ideal topologies.

math.LO↗
arXiv2026

Inquisitive first-order logic is neither compact nor recursively axiomatizable

Inquisitive first-order logic is an extension of classical first-order logic with formulas regimenting first-order questions, such as "whether all objects are P", "which objects are P", and "what is one object that is P". Since it was first developed in 2009, two major meta-theoretical questions about this logic have remained open, in spite of significant efforts. The first concerns compactness: if a conclusion follows from a set of premises, does it always follow from some finite subset? The second concerns the computational status of validity: is the set of validities recursively enumerable, or equivalently, does the logic admit a recursive axiomatization? We settle both questions in the negative, showing that inquisitive first-order logic is neither compact nor recursively axiomatizable. Furthermore, we prove that it violates another signature property of first-order logic, namely, Craig interpolation. We discuss the significance of our results, and show how to extend them to a closely related logic, viz., inquisitive team logic.

math.LO↗
arXiv2026

The Borel Distinguishing Number of Schreier Graphs

The Borel distinguishing number $D_B(\mathcal{G})$ of a Borel graph $\mathcal{G}$, recently introduced by Bilge and Kaya, is the minimum number of colors required to break the symmetry of $\mathcal{G}$ in a Borel way. In this paper, we investigate the Borel distinguishing number of Schreier graphs induced by the free part of the shift action $Γ\curvearrowright n^Γ$. We prove that $D_B(\mathcal{G})\le n+1$ for $Γ=\mathbb{Z}^d$ equipped with the standard generators. Moreover, we show that $D_B(\mathcal{G})\ge n+1$ if $ Γ$ is amenable and $ \{γ\in \mathrm{Aut}(\mathrm{Cay}(Γ,S)) \mid γ(e) = e \}$ is non-trivial. We also show that $D_B(\mathcal{G})$ is finite if $Γ$ is finitely generated, and give some applications of our results. These results answer some questions raised by Bilge and Kaya.

math.LO↗