Linkage of Sets of Cyclic Algebras

TL;DR

This paper constructs large collections of cyclic algebras over certain function fields that do not share any maximal subfield, highlighting differences based on the characteristic of the base field.

Contribution

It demonstrates the existence of many cyclic algebras with no common maximal subfield over function fields, depending on the characteristic of the base field.

Findings

For characteristic p ≥ 3, existence of p^2 - 1 cyclic algebras with no common maximal subfield.

For characteristic zero, existence of p^2 cyclic algebras with no common maximal subfield.

Shows characteristic-dependent differences in the linkage of cyclic algebras.

Abstract

Let $p$ be a prime integer and $F$ the function field in two algebraically independent variables over a smaller field $F_0$. We prove that if $\operatorname{char}(F_0)=p\geq 3$ then there exist $p^2-1$ cyclic algebras of degree $p$ over $F$ that have no maximal subfield in common, and if $\operatorname{char}(F_0)=0$ then there exist $p^2$ cyclic algebras of degree $p$ over $F$ that have no maximal subfield in common.