The Herbrand-Ribet theorem is a result in number theory concerning the class group of certain number fields. It strengthens an earlier theorem of Ernst Kummer, which states that a prime p divides the class number of the cyclotomic field of pth roots of unity if and only if p divides the numerator of some Bernoulli number with an even index below p minus 1. The Herbrand-Ribet theorem specifies more precisely what it means, within the structure of the class group, when p divides such a Bernoulli number's numerator. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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Statement FormCharacterization Theorem 1 Connections
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Source Herbrand-Ribet theorem (Wikipedia)
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Source Wikipedia: Herbrand-Ribet theorem
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1. Wikipedia: Herbrand-Ribet theorem
WikipediaLead section, statement-form reference
It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the n-th Bernoulli number Bn for some n, 0 < n < p − 1.
Proved By: Jacques Herbrand, Lead
The part saying p divides Bp−n if Gn is not trivial is due to Jacques Herbrand
View the Source Herbrand-Ribet theorem (Wikipedia)
Lead sectionQuote, Lead section
In mathematics, the Herbrand-Ribet theorem is a result on the class group of certain number fields.
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