Class number parity for cyclotomic fields

We give a simple criterion for the parity of the class number of the cyclotomic field.

     

On the first factor of the class number of prime cyclotomic fields

Let H(l) be the first factor of the class number of the field $\text{Q}$(exp 2πi/l), l a prime. The best-known upper and lower bounds on H(l) are improved for small l. The methods would also improve the best-known bounds for large l. It is shown that H(l) is the absolute value of the determinant of an easily written down matrix whose only entries are 0 and 1. The upper bounds obtained on H(l) significantly improve the Hadamard bound on the determinant of this matrix. Results of Lehmer on the factors of H(l) are explained via class field theory.     

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 .

     

The class number of cyclotomic function fields

Let k be a rational function field over a finite field. Carlitz and Hayes have described a family of extensions of k which are analogous to the collection of cyclotomic extensions {Q(ζm)| m ≥ 2} of the rational field Q. We investigate arithmetic properties of these “cyclotomic function fields.” We introduce the notion of the maximal real subfield of the cyclotomic function field and develop class number formulas for both the cyclotomic function field and its maximal real subfield. Our principal result is the analogue of a classical theorem of Kummer which for a prime p and positive integer n relates the class number of Q(ζpn + ζpn?1), the maximal real subfield of Q(ζpn), to the index of the group of cyclotomic units in the full unit group of Z[ζpn].     

Relative class number of imaginary Abelian fields of prime conductor below 10000

In this paper we compute the relative class number of all imaginary Abelian fields of prime conductor below 10000. Our approach is based on a novel multiple evaluation technique, and, assuming the ERH, it has a running time of , where is the conductor of the field.

     

On the parity of the class number of multiquadratic number fields

This paper investigates the 2-class group of real multiquadratic number fields. Let p₁,p₂,…,pn be distinct primes and . We draw a list of all fields K whose 2-class group is trivial.     

On the coefficients of Jacobi sums in prime cyclotomic fields

Let and be prime numbers, and let be a primitive root mod . For , denote by the Jacobi sum . We study the integers such that and . We give a list of properties that characterize these coefficients. Then we show some of their applications to the study of the arithmetic of , in particular to the study of Vandiver's conjecture. For , let be the number of distinct roots of in . We show that . We give closed formulas for the numbers and in terms of quadratic and cubic power residue symbols mod .

     

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

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

.     

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     

On the class number of p-th cyclotomic field

Let h^[-(p)h^-(p) be the relative class number of the p-th cyclotomic field. We show that logh^-(p) = [(p+3)/4] logp - [(p)/2] log2p+ log(1-b) + O(log₂² p)\log h^-(p) = {{p+3} \over {4}} \log p - {{p} \over {2}} \log 2\pi + \log (1-\beta ) + O(\log _2^2 p), where b\beta denotes a Siegel zero, if such a zero exists and p o -1 mod 4p\equiv -1\pmod {4}. Otherwise this term does not appear.     

Computations on the growth of the first factor for prime cyclotomic fields

Denote byh(p) the first factor of the class number of the prime cyclotomic fieldk(exp (2i/p)). The theorem:h(p ₂)>h(p ₁) if 641 p ₂>p ₁ 19 is proved by straightforward computation.     

The first factor of the class number of the p-th cyclotomic field

Kummer’s conjecture states that the relative class number of the p-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true—it cannot be true if we assume the truth of two other widely believed conjectures. We establish a new bound for the error term in Kummer’s conjecture, and more precisely we prove that ${\log(h_p^-)=\frac{p+3}{4} \log p +\frac{p}{2}\log(2\pi)+\log(1-\beta)+O(\log_2 p)}$ , where β is a possible Siegel zero of an ${L(s,\chi), \chi}$ odd.     

Computations on the growth of the first factor for prime cyclotomic fields II

It is shown by computing thath(p) — the first factor of the class number of the prime cyclotomic fieldk(exp(2i/p)) — is strictly increasing for 19p1097.