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Cyclotomic Fields and Zeta Values. Springer Monographs in Mathematics. Springer-Verlag. ISBN 3-540-33068-2. Zbl 1100.11002. Weisstein, Eric W. "Cyclotomic Field". MathWorld. "Cyclotomic field", Encyclopedia of Mathematics, EMS Press, 2001 [1994] On the Ring of Integers of Real Cyclotomic Fields.
n generate the group of cyclotomic units. If n is a composite number having two or more distinct prime factors, then ζ a n − 1 is a unit. The subgroup of cyclotomic units generated by (ζ a n − 1)/(ζ n − 1) with (a, n) = 1 is not of finite index in general. [2] The cyclotomic units satisfy distribution relations.
The values that a cyclotomic polynomial () may take for other integer values of x is strongly related with the multiplicative order modulo a prime number. More precisely, given a prime number p and an integer b coprime with p , the multiplicative order of b modulo p , is the smallest positive integer n such that p is a divisor of b n − 1 ...
Then the cyclotomic Euler system is the set of numbers ... Cyclotomic Fields and Zeta Values, Springer Monographs in Mathematics, Springer-Verlag, pp. ...
The zeta function of a curve over a finite field corresponds to a p-adic L-function. Weil's theorem relating the eigenvalues of Frobenius to the zeros of the zeta function of the curve corresponds to Iwasawa's main conjecture relating the action of the Iwasawa algebra on X to zeros of the p-adic zeta function.
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 () (岩澤 健吉), as part of the theory of cyclotomic fields.
Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field. That is, whenever an algebraic number field has a Galois group over Q that is an abelian group, the field is a subfield of a field obtained by adjoining a root of unity to the rational numbers.
The Dedekind zeta function of a number field, analogous to the Riemann zeta function, is an analytic object which describes the behavior of prime ideals in K. When K is an abelian extension of Q, Dedekind zeta functions are products of Dirichlet L-functions, with there being one factor for each Dirichlet character. The trivial character ...