Cyclotomic Number Fields Research Articles

On the class number of the lpth cyclotomic number field

The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick out some pairs (l, p) of two odd primes l and p such that the class number of each lpth cyclotomic number field is divisible by l even if p 1 (mod l). If p ≡ (mod l) for l ≥ 5 or p ≡ 1 (mod 32) for l = 3, then it is easy from the outset to achieve our intention of finding the factor l using the genus number formula. Otherwise it seems to be difficult. We wish to make it clear algebraically why the class number has the prime factor l.

On the stickelberger ideal of (2,...,2)-extensions of a cyclotomic number field

Let p be an odd prime number and n a natural number. Let K be a (2,...,2)-extension of the pn−th cyclotomic number field obtained by adjoining $$\sqrt {m_1 } $$ ,..., $$\sqrt {m_t } $$ , where m1,...,mt are rational integers. We get a system of generators of the minus part of the Stickelberger ideal of K, and calculate its index. This index is described as a product of some determinants of rational integer components. From this result, it is shown that the relative class number of K is expressed as this index.

On an index formula for the relative class number of a cyclotomic number field

With regard to the relative class number of a cyclotomic number field, the relation between Girstmair's formula and Maillet's determinant is explained.

The relative class numbers of certain imaginary abelian number fields and determinants

Let p be an odd prime. For a rational integer a, denote by R′( a) a rational integer that satisfies −(p − 1) 2 ≤ R′(a) ≤ (p − 1) 2 and R′( a) ≡ a (mod p). We define a determinant D′ p by D′ p = |R′(ab′)| 1 ≤ a, b ≤ (p − 1) 2 , where bb′ ≡ 1 (mod p). It is shown that D′ p is related to the relative class numbers of imaginary subfields of the 8 pth cyclotomic number field, and, if p ≡ ±3 (mod 8), D′ p ≠ 0. The other determinants are also studied.

On the unramified extensions of the prime cyclotomic number field and its quadratic extensions

It is interesting to know what kinds of primes are the factors of the class number of an algebraic number field, and especially to find ones being prime to the degree. About this matter it is desirable to construct the unramified Abelian extensions plainly. In this paper we shall show some of them for the prime cyclotomic number field and its quadratic extensions using the units of subfields.

On power bases in cyclotomic number fields

The current paper considers the question of power bases in the cyclotomic number field Q(ζ), ζ p = 1, p an odd prime. The ring of integers is Z[ζ], and there do exist further “non-obvious” generators for this order; specifically we shall see that Z[α] = Z[ζ] for α = ζ + ζ 2 + ⋯ + ζ (p−1) 2 . We conjecture that, up to conjugacy, there can be no further such integral generators, and prove that this is indeed the case in Q(ζ 7).

Cyclotomic polynomials and units in cyclotomic number fields

Let P( x) ≠ x be a monic irreducible polynomial with integer coefficients such that for infinitely many n, P(e 2πi n ) is a unit in the ring of algebraic integers Z[e 2πi n ] . It is shown that P( x) is a cyclotomic polynomial.

Class Numbers of Cyclotomic Function Fields

Using results of Galovich and Rosen the plus and minus factors for the class number of the cyclotomic function field associated with irreducibles of degree three and four over the-field with three elements are computed. As a consequence it is shown that the analogue to a result of Kummer on the p-divisibility of these factors is false. Consider an odd prime p and let F denote the finite field with p elements. To each monic irreducible polynomial P(T) of degree d with coefficients in F one can associate a cyclic extension Kp of F (T) that enjoys properties analogous to those of the classical cyclotomic number fields Q(tP), p = e2' /7. Such fields were dis- covered by L. Carlitz in the 1930's and have been studied intensively in recent years by D. Hayes, M. Rosen, S. Galovich, D. Goss and others. If D is the integral closure of F (T) in Kp, then one has the diagram:

Units and class groups in cyclotomic function fields

A unit index-class number formula is proven for “cyclotomic function fields” in analogy with similar results in cyclotomic number fields. The main theorem generalizes to arbitrary modules work done earlier for prime power modules.

The Hasse norm principle in cyclotomic number fields.

The Hasse norm principle in cyclotomic number fields.

Class numbers of algebraic number fields of Eisenstein type, II

Let K be an algebraic number field, of degree n, with a completely ramifying prime p, and let t be a common divisor of n and (p − 1) 2 . Then it is proved that if K does not contain the unique subfield, of degree t, of the p-th cyclotomic number field, then we have ( h K , n) > 1, where h K is the class number of K. As applications, we give several results on h K of such algebraic number fields K.

Units in cyclotomic number fields.

Units in cyclotomic number fields.