Class numbers of cyclotomic fields

In the first part of the paper we show how to construct real cyclotomic fields with large class numbers. If the GRH holds then the class number h_p⁺ of the pth real cyclotomic field satisfies h_p⁺ > p for the prime p = 11290018777. If we allow n to be composite we have, unconditionally, that $\text{h}{}_{\mathrm{n}}{}^{\mathrm{+}}\text{> n}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext><mtext>\; ?\; \epsilon </mtext>}}$ for infinitely many n. In the second part of the paper we show that if l ?= 2 mod 4 and n is the product of 4 distinct primes congruent to 1 mod l, then $\text{l}\text{2}$ (l, if l is odd) divides the class number h_n⁺ of the nth cyclotomic field. If the primes are congruent to 1 mod 4l then 2l divides h_n⁺.     

Class numbers of cyclotomic function fields

Let be a prime power and let be the finite field with elements. For each polynomial in , one could use the Carlitz module to construct an abelian extension of , called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of , similar to the role played by cyclotomic number fields for abelian extensions of . We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in . Two types of properties are obtained for the -parts of the class numbers of the fields in this tower, for a fixed prime number . One gives congruence relations between the -parts of these class numbers. The other gives lower bound for the -parts of these class numbers.

     

On the class numbers of cyclotomic fields

Let g_n denote the first factor of the class number of the nth cyclotomic field. It is proved that if n runs through a sequence of prime powers p^r tending to infinity, then

$\text{log}\text{g}{}_{\mathrm{n}}\text{～}\text{1}\text{4}\text{[1 ? (}\text{1}\text{p}\text{)]n}\text{log}\text{n}$

.     

Character coordinates and annihilators of cyclotomic numbers

The object of this paper is a representation theoretical approach to the problem of determining allQ-linear relations between conjugate numbers in a cyclotomic field. We apply our method to relations between the numbers cot^(m)(k/n), tan^(m)(k/n), cosec^(m)(2k/n), sec^(m)(2k/n), respectively, where m is0 and (k,n)=1. Thereby we complete previous work of Chowla, Hasse, Jager-Lenstra, and others.     

On the class numbers of cyclotomic fields

The finiteness of the number of cyclotomic fields whose relative class numbers have bounded odd parts will be verified and then all the cyclotomic fields with relative class numbers non-trivial 2-powers will be determined.This research was supported in part by Grant-in-Aid for Science (No. 01740051), Ministry of Education, Science, and Culture of Japan     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

Unique factorization in cyclotomic integers of degree seven

This article provides a survey of some basic results in algebraic number theory and applies this material to prove that the cyclotomic integers generated by a seventh root of unity are a unique factorization domain. Part of the proof uses the computer algebra system Maple to find and verify factorizations. The proofs use a combination of historic and modern techniques and some attempt has been made to discuss the history of this material.     

Graded decomposition numbers for cyclotomic Hecke algebras

In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero.     

Radical and cyclotomic extensions of the rational numbers

A radical extension of the rational numbers is a field generated by an element having a power in , and a cyclotomic extension is an extension generated by a root of unity. We show that a radical extension that is almost Galois over is almost cyclotomic. More precisely, we prove that if is radical with Galois closure , then contains a cyclotomic field such that the degree is bounded above by an almost linear function of . In particular, if is Galois, it contains a cyclotomic field such that .

     

Properties that characterize Gaussian periods and cyclotomic numbers

Let be a prime number, a -th primitive root of 1 and the periods of degree of . Write with . Several characterizations of the numbers and (or, equivalently, of the cyclotomic numbers of order ) are given in terms of systems of equations they satisfy and a condition on the linear independence, over , of the or on the irreducibility, over , of the characteristic polynomial of the matrix .

     

Class numbers of real cyclotomic fields of prime conductor

The class numbers of the real cyclotomic fields are notoriously hard to compute. Indeed, the number is not known for a single prime . In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields for the primes . It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue --on the basis of the Cohen-Lenstra heuristics-- that the probability that our table is actually a table of class numbers , is at least .

     

A specific family of cyclotomic polynomials of order three

Let A(n) be the largest absolute value of any coefficient of n-th cyclotomic polynomial Φn(x).We say Φn(x) is flat if A(n) = 1.In this paper,for odd primes p q r and 2r ≡ 1(mod pq),we prove that Φpqr(x) is flat if and only if p = 3 and q ≡ 1(mod 3).     

Unique expansions of real numbers

It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci∈[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type.     

Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Given an odd prime we show a way to construct large families of polynomials , , where is a set of primes of the form mod and is the irreducible polynomial of the Gaussian periods of degree in . Examples of these families when are worked in detail. We also show, given an integer and a prime mod , how to represent by matrices the Gaussian periods of degree in , and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of .

     

Gauss periods and codebooks from generalized cyclotomic sets of order four

Let p, q be distinct primes with gcd(p ? 1, q ? 1) = 4. Let D ₀, D ₁, D ₂, D ₃ be Whiteman’s generalized cyclotomic classes, satisfying the multiplicative group ${{\mathbb Z}^*_{pq}=D_0\cup D_1\cup D_2\cup D_3}$ . In this paper, we give formulas of Gauss periods: ${\sum_{i\in D_0\cup D_2}\zeta^i}$ and ${\sum_{i\in D_0}\zeta^i}$ , where ${\zeta}$ is a pqth primitive root of unity. As an application, we get the maximum cross-correlation amplitudes of three codebooks from generalized cyclotomic sets of order four and supply conditions on p and q such that they nearly meet the Welch bound.     

Unique representation bi-basis for rational numbers field

For \(A\subseteq {\mathbb {Q}}\), \(\alpha \in {\mathbb {Q}}\), let \(r_{A}(\alpha )=\#\{(a_{1}, a_{2})\in A^{2}: \alpha =a_{1}+a_{2}, a_{1}\le a_{2}\},\) \(\delta _{A}(\alpha )=\#\{(a_{1}, a_{2})\in A^{2}: \alpha =a_{1}-a_{2} \}.\) In this paper, we construct a set \(A\subset {\mathbb {Q}}\) such that \(r_{A}(\alpha )=1\) for all \(\alpha \in {\mathbb {Q}}\) and \(\delta _{A}(\alpha )=1\) for all \(\alpha \in {\mathbb {Q}}\setminus \{{0}\}\).     

广义Bernoulli数和广义高阶Bernoulli数

定义了广义Bernoulli数和广义高阶Bernoulli数，建立了它们的递推公式和有关性质，从而推广了Bernoulli数和高阶Bernoulli数。