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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 1493 times total.

Classification:
AMS MSC11R29 (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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