class number divisibility in p-extensions

In this entry, the class numberMathworldPlanetmathPlanetmath of a number fieldMathworldPlanetmath F is denoted by hF.

Theorem 1.

Let p be a fixed prime numberMathworldPlanetmath.

  • Let F/K be a Galois extensionMathworldPlanetmath with Galois groupMathworldPlanetmath Gal(F/K) and suppose F/K is a p-extensionPlanetmathPlanetmath (so Gal(F/K) is a p-group). Assume that there is at most one prime or archimedean place which ramifies in F/K. If hF is divisible by p then hK is also divisible by p.

  • Let F/ be a Galois extension of the rational numbers and assume that Gal(F/) is a p-group and at most one place (finite or infinite) ramifies then hF is not divisible by p.

Title class number divisibility in p-extensions
Canonical name ClassNumberDivisibilityInPextensions
Date of creation 2013-03-22 15:07:38
Last modified on 2013-03-22 15:07:38
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Theorem
Classification msc 11R29
Classification msc 11R37
Related topic PushDownTheoremOnClassNumbers
Related topic IdealClass
Related topic PExtension
Related topic ClassNumbersAndDiscriminantsTopicsOnClassGroups