PlanetMath: class number divisibility in $p$-extensions

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class number divisibility in $p$

-extensions

(Theorem)

In this entry, the class number of a number field is denoted by .

Theorem 1 Let be a fixed prime number.

Let be a Galois extension with Galois group $\operatorname{Gal}(F/K)$ and suppose is a -extension (so $\operatorname{Gal}(F/K)$ is a -group). Assume that there is at most one prime or archimedean place which ramifies in . If is divisible by then is also divisible by .
Let $F/\mathbb{Q}$ be a Galois extension of the rational numbers and assume that $\operatorname{Gal}(F/\mathbb{Q})$ is a -group and at most one place (finite or infinite) ramifies then is not divisible by .

"class number divisibility in $p$

-extensions" is owned by alozano.

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-extensions, born on 2005-03-10, modified 2006-06-10.
Object id is 6867, canonical name is ClassNumberDivisibilityInPExtensions.
Accessed 1008 times total.

Classification:

AMS MSC:	11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)
	11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory)

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