On the p-Rank of Class Groups of p-Extensions

Research output: Contribution to journalArticlepeer-review


We prove a local–global principle for the embedding problems of global fields with restricted ramification. By this local–global principle, for a global field k⁠, we use only the local information to give a presentation of the maximal pro-p Galois group of k with restricted ramification, when some Galois cohomological conditions are satisfied. For a Galois p-extension K/k⁠, we use our presentation result for k to study the structure of pro-p Galois groups of K⁠. Then for k=Q and k=Fq(t) with p∤q⁠, we give upper and lower bounds for the rank of p-torsion subgroup of the class group of K⁠, and these bounds depend on the structure of the Galois group and the inertia subgroups of K/k⁠. Finally, we study the p-rank of class groups of cyclic p-extensions of Q and the 2-rank of class groups of multiquadratic extensions of Q⁠, for a fixed ramification type.
Original languageEnglish (US)
Article numberrnad234
Pages (from-to)5274-5325
Number of pages52
JournalInternational Mathematics Research Notices
Issue number6
Early online dateOct 18 2023
StatePublished - Mar 1 2024


Dive into the research topics of 'On the p-Rank of Class Groups of p-Extensions'. Together they form a unique fingerprint.

Cite this