Cyclotomic Number Fields Research Articles

RAY CLASS GROUPS OF QUADRATIC AND CYCLOTOMIC FIELDS

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Let [Formula: see text] be a real quadratic number field with a prime P lying above p in ℚ. If p splits in K/ℚ and p does not divide the big class number of K, then any pro-p extension of K ramified only at P is finite cyclic. If p is inert in K/ℚ, then there exist infinite extensions of K ramified only at P. Furthermore, for big enough integer k, the ray class field (mod Pk+1) is obtained from the ray class field (mod Pk) by adjoining ζpk+1. In the case of a regular cyclotomic number field K = ℚ(ζp), the explicit structure of ray class groups (mod Pk) is given for any positive integer k, where P is the unique prime in K above p.

AN ELEMENTARY PROOF OF THE CONDUCTOR-DISCRIMINANT FORMULA

For a finite abelian extension K/ℚ, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the L-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of pn-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.

Ducci sequences and cyclotomic fields

Let G be an abelian group and . A Ducci sequence over G is a sequence of n-tuples , where . When G is finite, this sequence is eventually periodic. In this paper, we study Ducci sequences over , where , using properties of cyclotomic polynomials. We first characterize the vanishing sequences (i.e. those for which the cyclic part consists only of 0), as well as the tuples in the cyclic part of a sequence. We then derive an expression for the period of a given Ducci sequence in terms of orders of roots of unity plus one modulo powers of a prime above p in a cyclotomic number field. Lastly, the dependence of the period on t leads us to a connection with Wieferich primes.

Ring Isomorphism of Cyclic Cyclotomic Algebras

It is shown that ring isomorphisms between cyclic cyclotomic algebras over cyclotomic number fields are essentially determined by the list of local Schur indices at all rational primes. As a consequence, ring isomorphisms between simple components of the rational group algebras of finite metacyclic groups are determined by the center, the dimension over ℚ, and the list of local Schur indices at rational primes. An example is given to show that this does not hold for finite groups in general.

Ideal class groups of cyclotomic number fields III

Using an idea going back to Scholz, we construct unramified abelian extensions of cyclotomic extensions of number fields.

Maximal Diversity Algebraic Space–Time Codes With Low Peak-to-Mean Power Ratio

The design requirements for coding typically involves achieving the goals of good performance, high rates, and decoding complexity. In this paper, we introduce a further constraint on code design in that the code should also lead to values of the peak-to-mean envelope power ratio (PMEPR) for each antenna. Towards that end, we propose a new class of codes called the low PMEPR space-time (LPST) codes. The LPST codes are obtained using the properties of certain cyclotomic number fields. The LPST codes achieve a performance identical to that of the threaded algebraic (TAST) codes but at a much smaller PMEPR. With M antennas and a rate of one symbol per channel use, the LPST codes lead to a decrease in PMEPR by at least a factor of M relative to a Hadamard spread version of the TAST code. For rates beyond one symbol per channel use and up to a guaranteed amount, the LPST codes have provably smaller PMEPR than the corresponding TAST codes. Additionally, with the concept of punctured LPST codes proposed in this paper, significant performance improvement is obtained over the full diversity TAST schemes of comparable complexity. Numerical examples are provided to illustrate the advantage of the proposed codes in terms of PMEPR reduction and performance improvement for very high rate wireless communications.

Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups

A black-box secret sharing scheme for the threshold access structure T_t,n is one which works over any finite Abelian group G. Briefly, such a scheme differs from an ordinary linear secret sharing scheme (over, say, a given finite field) in that distribution matrix and reconstruction vectors are defined over Z and are designed independently of the group G from which the secret and the shares are sampled. This means that perfect completeness and perfect privacy are guaranteed regardless of which group G is chosen. We define the black-box secret sharing problem as the problem of devising, for an arbitrary given T_t,n, a scheme with minimal expansion factor, i.e., where the length of the full vector of shares divided by the number of players, n, is minimal. <br /> Such schemes are relevant for instance in the context of distributed cryptosystems based on groups with secret or hard to compute group order. A recent example is secure general multi-party computation over black-box rings. <br /> In 1994 Desmedt and Frankel have proposed an elegant approach to the black-box secret sharing problem based in part on polynomial interpolation over cyclotomic number fields. For arbitrary given T_t,n with 0 < t < n-1, the expansion factor of their scheme is O(n). This is the best previous general approach to the problem. <br /> Using low degree integral extensions of Z over which there exists a pair of sufficiently large Vandermonde matrices with co-prime determinants, we construct, for arbitrary given T_t,n with 0 < t < n-1, a black-box secret sharing scheme with expansion factor O(log n), which we show is minimal.

Computation ofL(0,χ) and of relative class numbers of CM-fields

Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number fieldLof discriminantdL. Then,L(0,χ) is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values ats= 0 of such abelian HeckeL-functions over totally real number fieldsL. Letfχdenote the norm of the finite part of the conductor ofχ. Then, roughly speaking, we can computeL(0,χ) inO((dLfx)0.5+∊) elementary operations. We then explain how the computation of relative class numbers of CM-fields boils down to the computation of exact values ats= 0 of such abelian HeckeL-functions over totally real number fieldsL. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations ofL(0,χ) over totally real number fields of degree 2 and 6.

Class number of (v, n, M)-extensions

An analogue of cyclotomic number fields for function fields over the finite field q, was investigated by L. Carlitz in 1935 and has been studied recently by D. Hayes, M. Rosen, S. Galovich and others. For each nonzero polynomial M in q [T], we denote by k (ΛM) the cyclotomic function field associated with M, where k = q(T). Replacing T by 1/T in k and considering the cyclotomic function field Fv that corresponds to (1/T)v+1 gets us an extension of k, denoted by Lv, which is the fixed field of Fv modulo . We define a (v, n, M)-extension to be the composite N = knk (Λm) Lv where kn is the constant field of degree n over k. In this paper we give analytic class number formulas for (v, n, M)-extensions when M has a nonzero constant term.

Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern

Let K be an algebraic number field with non-zero α , β ∈ K . Siegel showed in 1929 that there are only finitely many units ε 1 , ε 2 in K which satisfy the unit equation αε 1 + βε 2 =1. In this article we present a new algorithm for solving unit equations which utilizes methods from the geometry of numbers. For the fist time unit equations in number fields up to unit rank 10 and with more than 100,000 solutions are solved. By applying our algorithm to index form equations we compute all power integral bases in the cyclotomic number fields up to degree 12 and in Q ( ζ 17 ), Q ( ζ 19 ), Q ( ζ 23 ).

Two-primary algebraic K-theory of two-regular number fields

We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order.

Class numbers of cyclotomic function fields

Let q be a prime power and let Fq be the finite field with q elements. For each polynomial Q(T) in lFq[T], one could use the mo(lule to construct an abelian extension of 1Fq(T), called a cyclotomic extension. cyclotomic extensions play a fundamental role in the study of abelian extensions of 1Fq(T), similar to the role played by cyclotomic numrLber fields for abelian extensions of Q. We are interested in the tower of cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in 1Fq[T]. Two types of properties are obtained for the 1-parts of the class numbers of the fields in this tower, for a fixed prime number 1. One gives congruence relations between the I-parts of these class numbers. The otsher gives lower bound for the 1-parts of these class numbers. Systematic study of cyclotomic field extensions of rational numbers started in the nineteenth century with Kummer and was essential in his work on Fermat's Last Theorem. Towers of cyclotomic number fields were first investigated by Iwasawa in the mid 1950's. One major application of his theory is to determine the growth of the p-divisibility of the class numbers for the fields in the tower [1w]. The study of the cyclotomic theory of function fields started with [Ca] in 1930. Let p be a prime and let q be a power of p. regarded the rational function field k IFq (T) and the associated polynomial ring A -Fq [T] as analogs of the rational number field Q and its ring of integers Z. He constructed an Amodule, later called the Carlitz , out of the completion of the algebraic closure of IFq((T)), an analog of the field of complex numbers. For each polynomial P in A, one could use the module to construct a field extension k(P) of k. The extensions obtained this way are called cyclotomic extensions and are essential in the study of all abelian extensions of k. Fix an irreducible polynomial P in A, and let n run through the set of positive integers. The cyclotomic extensions of k associated to Pn via the module form a tower of extensions: nk c k(P) C k(p2) C .. C k(Pn) C It would be interesting to study the growth of the p-divisibility of the the class numbers for these fields along this tower, as Iwasawa did for cyclotomic extensions of a number field. This is the problem we would like to investigate in this paper. As in the case of a cyclotomic number field, one can decompose the class number h(k(Pn)) of k(Pn) into two integer factors h+(k(Pn)) and h-(k(PT)), called the real part and the relative part of the class number. Let p be the unique prime Received by the editors May 15, 1997. 1991 Mathematics Subject Classification. Primary 1IR29, 11R58; Secondary 11R23.

Ideal class groups of cyclotomic number fields II

Ideal class groups of cyclotomic number fields II

The Conductors of Eisenstein Characters in Cyclotomic Number Fields

The Conductors of Eisenstein Characters in Cyclotomic Number Fields

Ideal class groups of cyclotomic number fields I

Ideal class groups of cyclotomic number fields I

Modular invariants and fusion rule automorphisms from Galois theory

We show that Galois theory of cyclotomic number fields provides a powerful tool to construct systematically integer-valued matrices commuting with the modular matrix S, as well as automorphisms of the fusion rules. Both of these prescriptions allow the construction of modular invariants and offer new insight in the structure of known exceptional invariants.

Circular units of function fields

A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields.

The decomposition numbers of the hecke algebra of typeF 4

LetW be the finite Coxeter group of typeF 4, andH r (q) be the associated Hecke algebra, with parameter a prime powerq, defined over a valuation ringR in a large enough extension field ofQ, with residue class field of characteristicr. In this paper, ther-modular decomposition numbers ofH R (q) are determined for allq andr such thatr does not divideq. The methods of the proofs involve the study of the generic Hecke algebra of typeF 4 over the ringA = ℤ[u 1/2,u -1/2] of Laurent polynomials in an indeterminateu 1/2 and its specializations onto the ring of integers in various cyclotomic number fields. Substancial use of computers and computer program systems (GAP, MAPLE, Meat-Axe) has been made.

On the unramified Kummer extensions of quadratic extensions of the prime cyclotomic number field

On the unramified Kummer extensions of quadratic extensions of the prime cyclotomic number field

Splitting fields of association schemes

The splitting field K of a commutative association scheme is the extension of the rationals by the adjunction of all eigenvalues of the association scheme. Let L be a subfield of K containing all the Krein parameters. It is shown that the Galois group of K/L is contained in the center of the Galois group of K/ℚ. In particular, if the Krein parameters are all rational, then the eigenvalues are contained in a cyclotomic number field.