Herbrand–Ribet theorem

Herbrand–Ribet theorem

In mathematics, the Herbrand–Ribet theorem is a result on the class number of certain number fields. It is a strengthening of 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 nth Bernoulli number "B""n" for some "n", 0 < "n" < "p" − 1. The Herbrand–Ribet theorem specifies what, in particular, it means when "p" divides such an "B""n".

The Galois group &Sigma; of the cyclotomic field of "p"th roots of unity for an odd prime "p", Q(&zeta;) with &zeta;"p" = 1, consists of the "p" − 1 group elements &sigma;"a", where &sigma;"a" is defined by the fact that sigma_a(zeta) = zeta^a. As a consequence of the little Fermat theorem, in the ring of "p"-adic integers Bbb{Z}_p we have "p" − 1 roots of unity, each of which is congruent mod "p" to some number in the range 1 to "p" − 1; we can therefore define a Dirichlet character &omega; (the Teichmüller character) with values in Bbb{Z}_p by requiring that for "n" relatively prime to "p", &omega;("n") be congruent to "n" modulo "p". The "p" part of the class group is a Bbb{Z}_p-module, and we can apply elements in the group ring Bbb{Z}_p [Sigma] to it and obtain elements of the class group. We now may define an idempotent element of the group ring for each "n" from 1 to "p" − 1, as

:epsilon_n = frac{1}{p-1}sum_{a=1}^{p-1} omega(a)^n sigma_a^{-1}.

We now can break up the "p" part of the ideal class group "G" of Q(&zeta;) by means of the idempotents; if "G" is the ideal class group, then "G""n" = &epsilon;"n"("G").

Then we have the theorem of Herbrand–Ribet [Ribet, Ken, A modular construction of unramified p-extensions of Bbb{Q}(&mu;p), Inv. Math. 34 (3), 1976, pp. 151-162.] : "G""n" is nontrivial if and only if "p" divides the Bernoulli number "B""p"−"n". The part saying p divides "B""p"−"n" if "G""n" is not trivial is due to Herbrand. The converse, that if "p" divides "B""p"−"n" then "G""n" is not trivial is due to Ribet, and is considerably more difficult. By class field theory, this can only be true if there is an unramified extension of the field of "p"th roots of unity by a cyclic extension of degree "p" which behaves in the specified way under the action of &Sigma;; Ribet proves this by actually constructing such an extension using methods in the theory of modular forms. A more elementary proof of Ribet's converse to Herbrand's theorem can be found in Washington's book. [Washington, Lawrence C., Introduction to Cyclotomic Fields, Second Edition, Springer-Verlag, 1997.]

Ribet's methods were pushed further by Barry Mazur and Andrew Wiles in order to prove the Main Conjecture of Iwasawa Theory, [Mazur, Barry, and Wiles, Andrew, Class Fields of Abelian Extension of Bbb{Q}, Inv. Math. 76 (2), 1984, pp. 179-330.] a corollary of which is a strengthening of the Herbrand-Ribet theorem: the power of "p" dividing "B""p"−"n" is exactly the power of "p" dividing the order of "G""n".

ee also

*Iwasawa Theory

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Théorème de Herbrand-Ribet — Le théorème de Herbrand Ribet est un renforcement du théorème de Kummer avec pour effet le fait que le nombre premier p divise le nombre de classes du corps cyclotomique des racines p ièmes de l unité si et seulement si p divise le numérateur du… …   Wikipédia en Français

  • Jacques Herbrand — (February 12, 1908 July 27, 1931) was a French mathematician who was born in Paris, France and died in La Bérarde, Isère, France. He worked in mathematical logic and class field theory. He introduced recursive functions. Herbrand s theorem refers …   Wikipedia

  • Kenneth Alan Ribet — Kenneth Alan Ken Ribet is an American mathematician, currently a professor of mathematics at the University of California, Berkeley. His mathematical interests include algebraic number theory and algebraic geometry.He is credited with paving the… …   Wikipedia

  • Kenneth Alan Ribet — Ken Ribet Ken Ribet en 2007 au CIRM Naissance 28 juin 1948 (États Unis) Domicile États Unis Nationalité …   Wikipédia en Français

  • List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia

  • Iwasawa theory — In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of …   Wikipedia

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

  • Bernoulli number — In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers. There are several conventions for… …   Wikipedia

  • List of algebraic number theory topics — This is a list of algebraic number theory topics. Contents 1 Basic topics 2 Important problems 3 General aspects 4 Class field theory …   Wikipedia

  • Regular prime — In number theory, a regular prime is a certain kind of prime number. A prime number p is called regular if it does not divide the class number of the p th cyclotomic field (that is, the algebraic number field obtained by adjoining the p th root… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”