class number divisibility in cyclic extensions
In this entry, the class number of a number field is denoted by .
Theorem 1.
Let be a cyclic Galois extension of degree . Let be a prime such that is not divisible by , and assume that does not divide , the class number of any intermediate field . If divides then also divides , where is the multiplicative order of modulo .
Recall that the multiplicative order of modulo is a number such that and is not congruent to modulo for any .
Corollary 1.
Let be a Galois extension such that is a prime distinct from the prime . Assume that does not divide . If divides then divides , where is the multiplicative order of modulo .
Note that a Galois extension of prime degree has no non-trivial subextensions.
Title | class number divisibility in cyclic extensions |
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Canonical name | ClassNumberDivisibilityInCyclicExtensions |
Date of creation | 2013-03-22 15:07:41 |
Last modified on | 2013-03-22 15:07:41 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 4 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 11R29 |
Classification | msc 11R32 |
Classification | msc 11R37 |
Related topic | IdealClass |
Related topic | ClassNumbersAndDiscriminantsTopicsOnClassGroups |