Abstract
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 language | English (US) |
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Article number | rnad234 |
Pages (from-to) | 5274-5325 |
Number of pages | 52 |
Journal | International Mathematics Research Notices |
Volume | 2024 |
Issue number | 6 |
Early online date | Oct 18 2023 |
DOIs | |
State | Published - Mar 1 2024 |