push-down theorem on class numbers
As in the parent (http://planetmath.org/ClassNumberDivisibilityInExtensions) entry, given a number field^{} $K$, the class number^{} of $K$ is denoted by ${h}_{K}$.
Theorem (Pushing-Down Theorem).
Let $E\mathrm{/}F$ be a $p$-extension of number fields and suppose that only one prime ideal^{} of $F$ is ramified in $E$ and that this prime is totally ramified. Then $p\mathrm{|}{h}_{E}$ implies $p\mathrm{|}{h}_{F}$.
References
- Fröh A. Fröhlich, On a method for the determination of class number factors in number fields, Mathematika, 4 (1957), 113-121.
- Iwas K. Iwasawa, A note on Class Numbers of Algebraic Number Fields, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 257-258.
Title | push-down theorem on class numbers |
Canonical name | PushdownTheoremOnClassNumbers |
Date of creation | 2013-03-22 15:05:19 |
Last modified on | 2013-03-22 15:05:19 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 6 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 11R37 |
Classification | msc 11R32 |
Classification | msc 11R29 |
Related topic | IdealClass |
Related topic | PExtension |
Related topic | ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility |
Related topic | ClassNumberDivisibilityInPExtensions |
Related topic | ClassNumbersAndDiscriminantsTopicsOnClassGroups |