PlanetMath: class number divisibility in $p$-extensions

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

class number divisibility in $p$

-extensions

(Theorem)

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 $\Gal(F/K)$ and suppose $F/K$ is a $p$ extension (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 $h_F$ is divisible by $p$ then $h_K$ is also divisible by $p$
Let $F/\Rats$ be a Galois extension of the rational numbers and assume that $\Gal(F/\Rats)$ is a $p$ group and at most one place (finite or infinite) ramifies then $h_F$ is not divisible by $p$

"class number divisibility in $p$

-extensions" is owned by alozano.

(view preamble | get metadata)

Log in to rate this entry.
(view current ratings)

-extensions, born on 2005-03-10, modified 2006-06-10.
Object id is 6867, canonical name is ClassNumberDivisibilityInPExtensions.
Accessed 1382 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)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact