# 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 $E/F$ 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 $p|h_{E}$ implies $p|h_{F}$.

## 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