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[parent] push-down theorem on class numbers (Theorem)

As in the parent entry, given a number field $ K$, the class number of $ K$ is denoted by $ h_K$.

Theorem 1 (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\vert h_E$ implies $ p\vert h_F$.

Bibliography

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.



"push-down theorem on class numbers" is owned by alozano.
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See Also: ideal class, $p$-extension, extensions without unramified subextensions and class number divisibility, class number divisibility in $p$-extensions, topics on ideal class groups and discriminants


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Cross-references: implies, totally ramified, prime, ramified, prime ideal, class number, number field
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This is version 3 of push-down theorem on class numbers, born on 2005-02-23, modified 2005-02-23.
Object id is 6815, canonical name is PushDownTheoremOnClassNumbers.
Accessed 1101 times total.

Classification:
AMS MSC: 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)
 11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory)
 11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory)
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