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[parent] class number divisibility in cyclic extensions (Theorem)

In this entry, the class number of a number field $ L$ is denoted by $ h_L$.

Theorem 1   Let $ F/K$ be a cyclic Galois extension of degree $ n$. Let $ p$ be a prime such that $ n$ is not divisible by $ p$, and assume that $ p$ does not divide $ h_E$, the class number of any intermediate field $ K\subseteq E \subsetneq F$. If $ p$ divides $ h_F$ then $ p^f$ also divides $ h_F$, where $ f$ is the multiplicative order of $ p$ modulo $ n$.

Recall that the multiplicative order of $ p$ modulo $ n$ is a number $ f$ such that $ p^f\equiv 1 \mod n$ and $ p^m$ is not congruent to $ 1$ modulo $ n$ for any $ 1\leq m <f$.

Corollary 1   Let $ F/K$ be a Galois extension such that $ [F:K]=q$ is a prime distinct from the prime $ p$. Assume that $ p$ does not divide $ h_K$. If $ p$ divides $ h_F$ then $ p^f$ divides $ h_F$, where $ f$ is the multiplicative order of $ p$ modulo $ q$.

Note that a Galois extension $ F/K$ of prime degree has no non-trivial subextensions.



"class number divisibility in cyclic extensions" is owned by alozano.
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See Also: ideal class, topics on ideal class groups and discriminants


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Cross-references: congruent, number, multiplicative order, field, divide, divisible, prime, degree, Galois extension, cyclic, number field, class number
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This is version 1 of class number divisibility in cyclic extensions, born on 2005-03-10.
Object id is 6868, canonical name is ClassNumberDivisibilityInCyclicExtensions.
Accessed 1245 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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