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class number divisibility in cyclic extensions
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(Theorem)
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In this entry, the class number of a number field $L$ is denoted by $h_L$ .
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.
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"class number divisibility in cyclic extensions" is owned by alozano.
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Cross-references: congruent, number, multiplicative order, field, divide, divisible, prime, degree, Galois extension, cyclic, number field, class number
There is 1 reference to this entry.
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 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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Pending Errata and Addenda
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