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.

• Let $F/K$ be a Galois extension with Galois group $\operatorname{Gal}(F/K)$ and suppose $F/K$ is a $p$-extension (so $\operatorname{Gal}(F/K)$ 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 $p$.

• Let $F/\mathbb{Q}$ be a Galois extension of the rational numbers and assume that $\operatorname{Gal}(F/\mathbb{Q})$ 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 ClassNumberDivisibilityInPextensions 2013-03-22 15:07:38 2013-03-22 15:07:38 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 11R29 msc 11R37 PushDownTheoremOnClassNumbers IdealClass PExtension ClassNumbersAndDiscriminantsTopicsOnClassGroups