push-down theorem on class numbers


As in the parent (http://planetmath.org/ClassNumberDivisibilityInExtensions) entry, given a number fieldMathworldPlanetmath K, the class numberMathworldPlanetmathPlanetmath of K is denoted by hK.

Theorem (Pushing-Down Theorem).

Let E/F be a p-extension of number fields and suppose that only one prime idealPlanetmathPlanetmathPlanetmath of F is ramified in E and that this prime is totally ramified. Then p|hE implies p|hF.

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