Class Numbers Of Cyclotomic Fields Research Articles

Norm relations and computational problems in number fields

For a finite group G $G$ , we introduce a generalization of norm relations in the group algebra Q [ G ] $\mathbb {Q}[G]$ . We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arithmetic invariants of the subfields of a normal extension of algebraic number fields with Galois group G $G$ . On the algorithmic side, this leads to subfield based algorithms for computing rings of integers, S $S$ -unit groups and class groups. For the S $S$ -unit group computation this yields a polynomial time reduction to the corresponding problem in subfields. We compute class groups of large number fields under GRH, and new unconditional values of class numbers of cyclotomic fields.

Real cyclotomic fields of prime conductor and their class numbers

Surprisingly, the class numbers of cyclotomic fields have only been determined for fields of small conductor, e.g., for prime conductors up to 67, due to the problem of finding the “plus part,” i.e., the class number of the maximal real subfield. Our results have improved the situation. We prove that the plus part of the class number is 1 for prime conductors between 71 and 151. Also, under the assumption of the generalized Riemann hypothesis, we determine the class number for prime conductors between 167 and 241. This technique generalizes to any totally real field of moderately large discriminant, allowing us to confront a large class of number fields not previously treatable.

On generalized Mersenne Primes and class-numbers of equivalent quadratic fields and cyclotomic fields

In this paper we define equivalent quadratic fields and prove that generalized Mersenne primes generate a family of infinitely many equivalent quadratic fields with equivalent index \(2\) and whose class numbers are divisible by 3. We also prove that the class-number of the cyclotomic field \(\mathbb {Q}\big ( \zeta _m \big )\), where \(m\in \mathbb {N}\) and \(\zeta _m\) is a primitive \(m\)-th root of unity, is divisible by a certain integer \(g\).

Upper bounds on relative class numbers of cyclotomic fields

Abstract We explain how one can use the explicit formulas for the mean square values of L-functions which we established elsewhere to obtain explcit upper bounds on relative class numbers of cyclotomic number fields. As an example, we show that the relative class numbers of the cyclotomic fields of conductor 4p, p ≥ 3 a prime, are less than or equal to 8√p(p/16)(p−1)/2.

Class numbers of real cyclotomic fields of composite conductor

AbstractUntil recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.

On the 2-part of the class numbers of cyclotomic fields of prime power conductors

Let p be an odd prime number and l a prime number with l ≠ p. Let Kn = Q(ζpn+1) be the pn+1-st cyclotomic field. Let hn and hn- be the class number and the relative class number of Kn, respectively. When l = 2, we give an explicit bound mp depending on p such that the ratio hn-/hn-1- is odd for all n > mp. When l ≥ 3, we also give a corresponding result on the l-part of the relative class number of Kn+(ζl). As an application, we show that when p ≤ 509, the ratio hn/h0 is odd for all n ≥ 1.

Mean values of L-functions and relative class numbers of cyclotomic fields

Using formulas for quadratic mean values of L-functions at s = 1, we recover previously known explicit upper bounds on relative class numbers of cyclotomic elds. We also obtain new better bounds.

A GENERALISED KUMMER'S CONJECTURE

AbstractKummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott–Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.

Prime factors of class number of cyclotomic fields

Soit p un nombre premier impair, r une racine primitive modulo p et r i ≡ r i (mod p) avec 1 < r i < p - 1. En 2007, R. Queme a pose la question: le l-rang (l premier impair ≠ p) du groupe des classes d'ideaux du p-ieme corps cyclotomique est-il egal au degre du plus grand diviseur commun sur le corps fini F l de x (p-1)/2 +1 et du polynome de Kummer f(x) = Σ p-2 i=0 r- i x i . Dans cet article, nous donnons une reponse complete a cette question en produisant un contre-exemple.

The mean values of Dirichlet L-functions at integer points and class numbers of cyclotomic fields

Let q be a positive integer, and L(s, χ) the Dirichlet L-function corresponding to a Dirichlet character χ mod q. We putwhere χ runs over all Dirichlet characters mod q except for the principal character χ0.

Conjectures in arithmetic algebraic geometry: a survey

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

Computation of the first factor of the class number of cyclotomic fields

We show how to compute the values of h 1( p), the first factor of the class number of the cyclotomic field Q (exp 2iπ/ p), for each prime p≤3000, and determine the set of prime divisors for each p≤1000. We confirm, for these values, a number of well known conjectures about h 1( p). We give some reasons why we believe that Kummer's conjectured asymptotic estimate for h 1( p) is likely to be wrong. We show how an extension of the recent work of Goldfeld, Cross, and Zagier might be used to establish that h 1( p) is monotone increasing for all p≥19.

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.

On the Stickelberger Ideal and the Relative Class Number

Let $k$ be any imaginary abelian field, $R$ the integral group ring of $G = {\text {Gal}}(k/\mathbb {Q})$, and $S$ the Stickelberger ideal of $k$. Roughly speaking, the relative class number ${h^ - }$ of $k$ is expressed as the index of $S$ in a certain ideal $A$ of $R$ described by means of $G$ and the complex conjugation of $k;{c^ - }{h^ - } = [A:S]$, with a rational number ${c^ - }$ in $\frac {1} {2}\mathbb {N} = \{ n/2;n \in \mathbb {N}\}$, which can be described without ${h^ - }$ and is of lower than ${h^ - }$ if the conductor of $k$ is sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that $2{c^ - }$, a natural number, divides $2{([k:\mathbb {Q}]/2)^{[k:\mathbb {Q}]/2}}$. In particular, if $k$ varies through a sequence of imaginary abelian fields of degrees bounded, then ${c^ - }$ takes only a finite number of values. On the other hand, it will be shown that ${c^ - }$ can take any value in $\frac {1} {2}\mathbb {N}$ when $k$ ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.

On the Stickelberger ideal and the relative class number

Let k k be any imaginary abelian field, R R the integral group ring of G = Gal ( k / Q ) G = {\text {Gal}}(k/\mathbb {Q}) , and S S the Stickelberger ideal of k k . Roughly speaking, the relative class number h − {h^ - } of k k is expressed as the index of S S in a certain ideal A A of R R described by means of G G and the complex conjugation of k ; c − h − = [ A : S ] k;{c^ - }{h^ - } = [A:S] , with a rational number c − {c^ - } in 1 2 N = { n / 2 ; n ∈ N } \frac {1} {2}\mathbb {N} = \{ n/2;n \in \mathbb {N}\} , which can be described without h − {h^ - } and is of lower than h − {h^ - } if the conductor of k k is sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that 2 c − 2{c^ - } , a natural number, divides 2 ( [ k : Q ] / 2 ) [ k : Q ] / 2 2{([k:\mathbb {Q}]/2)^{[k:\mathbb {Q}]/2}} . In particular, if k k varies through a sequence of imaginary abelian fields of degrees bounded, then c − {c^ - } takes only a finite number of values. On the other hand, it will be shown that c − {c^ - } can take any value in 1 2 N \frac {1} {2}\mathbb {N} when k k ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.

The 2-divisibility of the class number of cyclotomic fields and the Stickelberger ideal.

The 2-divisibility of the class number of cyclotomic fields and the Stickelberger ideal.

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 h n + > n 3 2 − ε 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 l 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 4 l then 2 l divides h n +.

On the Stickelberger ideal and the relative class number

Let k be any imaginary abelian fieldX R the integral group ring of G = Gal(k/(2) and S the Stickelberger ideal of k. Roughly speakingX the relative class number hof k is expressed as the index of S in a certain ideal A of R described by means of G and the complex conjugation of k; c-h= [A: S] with a rational number cin -NJ = {n/2;n e NJ} which can be described without hand is of lower than hif the conductor of k is sufficiently large (cf. [6 9 1O]; see also [5]). We shall prove that 2ca natural numberX divides 2([k: (2]/2)lk 1/2. In particularX if k varies through a sequence of imaginary abelian fields of degrees boundedX then ctakes only a finite number of values. On the other handX it will be shown that ccan take any value in 2NJ when k ranges over all imaginary abelian fields. In this connectionX we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields. Let 2, C;!!, 1R, and C denote the rational integer ring, the rational number field, the real number field, and the complex number field, respectively. A finite abelian extension over C;!! contained in C will be called an abelian field. Let k be an imagi- nary abelian field, namely, an abelian field not contained in 1R. We denote by R(k) the group ring of the Galois group G = Gal(k/C;!!) over z and by s(H), for any subgroup H of G, the sum in R(k) of all elements in H. Put A(k) = { E R(k); (1 + jk)°l = as(G) for some a E E}, where ik denotes the complex conjugation of k. Let hk denote the relative class number of k (i.e., the so-called first factor of the class number of k), Qk the unit index of k, 9k the number of distinct prime numbers ramified in k, and S(k) the Stickelberger ideal of k in the sense of Iwasawa-Sinnott, which is an additive sub- group of A(k) with finite index (for the definition of the Stickelberger ideal, see [6, 10]). We define ck as the ratio of the index [A(k): S(k)] to hk: Ck hk = [A(k): S(k)] The product QkCk is known to be a natural number and is determined by Sinnott in various cases, for example, in the case 9k = 1 or 2 (cf. [10]). He has also shown in [9] that, if k is a cyclotomic field, then Ck = 2b where b = 0 or 29k-1-1 according as 9k = 1 or 9k > 2 (for the case 9k = 1, see [6]). In this paper, we shall give an additional result concerning the range of Ck . Received by the editors July 31 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary llR20 llR29; Secondary llN25 llR18.

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 log g n ∼ 1 4 [1 − ( 1 p )]n log n .