Unification of Chowla’s Problem and Maillet–Demyanenko Determinants

First, we prove that a non-abelian normal CM-field of degree 16 has odd relative class number if and only if it is dihedral, or is a compositum of two normal octic CM-fields with the same maximal real subfield, or has Galois group . Then, we solve several relative class number one problems. (1) We solve the relative class number one problem for the dihedral CM-fields of 2-power degrees. First, we remind the reader of the characterization of the dihedral CM-fields of 2-power degrees with odd relative class numbers. Second, we give lower bounds on relative class numbers of dihedral CM-fields of 2-power degrees with odd relative class numbers. We thus obtain an upper bound on the discriminants of the dihedral CM-fields of 2-power degrees with relative class number equal to 1. Third, we compute the relative class numbers of all the dihedral CM-fields of 2-power degrees with odd relative class numbers and discriminants less than or equal to this latter bound. We end up with a list of twenty-four dihedral CM-fields of 2-power degrees with relativeclass numbers equal to 1, and show that exactly twenty-one of them have class number 1. (2) We determine all the non-abelian normal CM-fields of degree 16 with Galois group which have relative class number 1 (there is only one such number field), and then those which have class number 1 (there is only one such number field). (3) We determine some of the non-abelian normal CM-fields with the same maximal real subfield which have relative class number 1, and then those which have class number 1. Indeed, we focus on the case where one of the is a quaternion octic CM-field and prove that there is only one such compositum with relative class number 1 and that this compositum has class number 1.1991 Mathematics Subject Classification: primary 11R29; secondary 11R21, 11R42, 11M20, 11Y40.

We will show that the normal CM-fields with relative class number one are of degrees ≤ 216 \leq 216 . Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees ≤ 96 \leq 96 , and the CM-fields with class number one are of degrees ≤ 104 \leq 104 . By many authors all normal CM-fields of degrees ≤ 96 \leq 96 with class number one are known except for the possible fields of degree 64 64 or 96 96 . Consequently the class number one problem for normal CM-fields is solved under the Generalized Riemann Hypothesis except for these two cases.

The relative class number <TEX>$H_d(f)$</TEX> of a real quadratic field <TEX>$K=\mathbb{Q}(\sqrt{m})$</TEX> of discriminant d is the ratio of class numbers of <TEX>$O_f$</TEX> and <TEX>$O_K$</TEX>, where <TEX>$O_K$</TEX> denotes the ring of integers of K and <TEX>$O_f$</TEX> is the order of conductor f given by <TEX>$\mathbb{Z}+fO_K$</TEX>. In a recent paper of A. Furness and E. A. Parker the relative class number of <TEX>$\mathbb{Q}(\sqrt{m})$</TEX> has been investigated using continued fraction in the special case when <TEX>$(\sqrt{m})$</TEX> has a diagonal form. Here, we extend their result and show that there exists a conductor f of relative class number 1 when the continued fraction of <TEX>$(\sqrt{m})$</TEX> is non-diagonal of period 4 or 5. We also show that there exist infinitely many real quadratic fields with any power of 2 as relative class number if there are infinitely many Mersenne primes.

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.

Article Unit signatures, and even class numbers, and relative class numbers. was published on January 1, 1975 in the journal Journal für die reine und angewandte Mathematik (volume 1975, issue 274-275).

Let N be an imaginary abelian number field. We know that h-, the relative class number of N, goes to infinity as fN, the conductor of N, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CMfields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Second, we have proved in this paper that there are exactly 48 such fields.

A Demjanenko Matrix for Abelian Fields of Prime Power Conductor

We give a small, but very useful modification of a criterion of Mignotte ([4]) for Catalan’s equation, replacing the class number of a certain abelian field by the relative class number, which is much easier to compute. The proof is the same, apart from the idea to consider the class group modulo the ideals coming from the real subfield. We use the following notation: K is a CM-field, IK its group of fractional ideals and i : K∗ → IK the canonical map x 7→ (x); j denotes complex conjugation, K+ the maximal real subfield and h−(K) the relative class number of K; OK is the ring of integral elements of K. Lemma 1. Let K be a CM-field and Q a finite set of prime ideals of K. There is a subgroup I0 of the ideal group IK such that (i) the prime ideals in Q do not appear in the factorization of any ideal in I0; (ii) IK/(i(K∗)I0) has cardinality h−(K) or 2h−(K); (iii) if e ∈ K∗ with (e) ∈ I0, then e1−j is a root of unity. P r o o f. Let I0 consist of those ideals which are in the image of the canonical map IK+ → IK , and which do not contain any prime ideal in Q. If (e) ∈ I0, then (e) = (e), so e1−j is a unit, hence also a root of unity because all its conjugates have absolute value 1 (cf. [6], Lemma 1.6). It remains to show (ii). It is an easy consequence of the approximation theorem that every ideal class contains an ideal without primes in Q (see e.g. [3], IV, Corollary 1.4). Therefore IK/(i(K∗)I0) = ideal class group of K modulo image of the ideal class group of K+. By [6], Theorem 10.3, at most 2 ideal classes of K+ become principal in K, so the statement follows. Theorem 1. Let p 6= q be odd prime numbers. Let ζ be a primitive p-th root of unity and K an imaginary subfield of L := Q(ζ). Catalan’s equation x − y = 1 has no nontrivial integral solution if q -h−(K) and pq−1 6≡ 1 mod q2.

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.

Let F F be a finite field, A = F [ T ] A=F[T] , and k = F ( T ) k=F(T) . Let K m = k ( Λ m ) K_{m}=k(\Lambda _{m}) be the field extension of k k obtained by adjoining the m m -torsion on the Carlitz module. The class number h m h_{m} of K m K_{m} can be written as a product h m = h m + h m − h_{m}=h_{m}^{+}h_{m}^{-} . The number h m − h_{m}^{-} is called the relative class number. In this paper a formula for h m − h_{m}^{-} is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of p p -th roots of unity. Some consequences of this formula are also derived.

1. Introduction. It is known (cf. [1], [3], [4], [5], [7]) that there exist infinitely many imaginary quadratic fields each with class number divisible by a given integer. In this paper we show another simple proof of the above theorem. By a similar method we also construct infinitely many imaginary cyclic quartic fields whose relative class numbers are divisible by l, where l is a given prime congruent to 1 modulo 4. Further we characterize the imaginary quadratic fields whose class numbers are multiples of a given prime, by describing them explicitly. In the following, for an integer :r in an algebraic number field K we denote by (~) the principal ideal in K generated by ~. We write p ]] n for a prime p and an

formula for the ratio of the class number of a quadratic integral domain in a real field to the class number of the whole integral domain (of all quadratic integers in that field), with the principal objective of showing that this ratio takes many values (such as 1) infinitely often for the real case, in support of a conjecture of Gauss. The object of this paper is first of all to give Dirichlet's results briefly, together with some theorems and illustrations immediately deducible from them (in order to restrict the computation to cases in which the theory is of more help). We shall, of course, offer various tables of relative class numbers, such data being our main object. We emphasize quadratic integral domains of prime power conductor under the whole integral domain (of all quadratic integers of the field). We ask, in particular, when the relative class number is divisible by 2 and 4, and find simple linear congruence conditions. When we ask which prime conductors have relative class numbers divisible by 3, we find such primes are essentially the splitting primes of certain cubic fields and therefore representable by quadratic forms, according to the classic work of Dedekind [3]. This is basically an application of class-field theory and perhaps the tables emerging would be of some experimental use. The classic background is amplified in [7], [5], and [2]. Here it might be appropriate to remark that the tables given below have a natural limit of diminishing returns owing to the fact that the relevant portions of classical algebraic number theory were developed long ago with relatively little data, and it would be desirable to see the theory profit from more data before great feats of computer endurance are attempted.

We give a brief survey of three papers of K. Ramachandra in algebraic number theory. The first paper is based on his thesis and appeared in the Annals of Mathematics and titled, ``Some Applications of Kronecker's Limit Formula.'' The second paper determines a system of fundamental units for the cyclotomic field and is titled, ``On the units of cyclotomic fields.'' This appeared in Acta Arithmetica. The third deals with relative class numbers and is titled, ``The class number of relative abelian fields.'' This appeared in Crelle's Journal.

Calculating formula and divisibility for relative class numbers of abelian function fields

Demjanenko matrix and recursion formula for relative class number over function fields