class number divisibility in $p$extensions
In this entry, the class number^{} of a number field^{} $F$ is denoted by ${h}_{F}$.
Theorem 1.
Let $p$ be a fixed prime number^{}.

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Let $F/K$ be a Galois extension^{} with Galois group^{} $\mathrm{Gal}(F/K)$ and suppose $F/K$ is a $p$extension^{} (so $\mathrm{Gal}(F/K)$ is a $p$group). Assume that there is at most one prime or archimedean place which ramifies in $F/K$. If ${h}_{F}$ is divisible by $p$ then ${h}_{K}$ is also divisible by $p$.

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Let $F/\mathbb{Q}$ be a Galois extension of the rational numbers and assume that $\mathrm{Gal}(F/\mathbb{Q})$ is a $p$group and at most one place (finite or infinite) ramifies then ${h}_{F}$ is not divisible by $p$.
Title  class number divisibility in $p$extensions 

Canonical name  ClassNumberDivisibilityInPextensions 
Date of creation  20130322 15:07:38 
Last modified on  20130322 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 