Sylow-conjugate number fields

TL;DR

This paper demonstrates that a number field cannot be uniquely identified solely by the structure of the Sylow subgroups of its absolute Galois group, challenging previous assumptions about field determination.

Contribution

It provides a counterexample showing that Sylow subgroup structures do not determine a number field, answering a question posed by Florian Pop.

Findings

Number fields are not uniquely determined by Sylow subgroups of their Galois groups.

Counterexamples exist where different fields share identical Sylow subgroup structures.

The result contrasts with the classical Neukirch-Uchida theorem linking fields to their Galois groups.

Abstract

By a classical result of Neukirch and Uchida, a number field K is determined by the structure of its absolute Galois group Gal(K). We show that K is not determined by the structure of the Sylow subgroups of Gal(K), answering a question raised by Florian Pop.