zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Diophantine equations and class numbers. (English) Zbl 0591.12006
In this paper, the author intends to extend some already known results. First he tries to provide sufficient conditions for non-trivial class numbers of real quadratic fields of Richaud-Degert type, in order to generalize results of Ankeny-Chowla-Hasse, S. D. Lang, H. Takeuchi and I. Yamaguchi. However, almost all of them essentially have been already obtained by {\it H. Hasse} [Elem. Math. 20, 49-59 (1965; Zbl 0128.035)] and the reviewer [Nagoya Math. J. 91, 151-161 (1983; Zbl 0506.10012)]. Next he obtains necessary and sufficient conditions for an algebraic integer, not a unit, to be the norm of an algebraic integer in a given extension field. By using this, finally, he gives sufficient conditions for the divisibility of the class numbers of imaginary quadratic fields ${\bbfQ}(\sqrt{n\sp 2-a\sp t})$ by t, which are a generalization of works of {\it M. J. Cowles} [J. Number Theory 12, 113-115 (1980; Zbl 0427.12001)] and {\it B. H. Gross} and {\it D. E. Rohrlich} [Invent. Math. 44, 201-224 (1978; Zbl 0369.14011)].
Reviewer: H.Yokoi

11R11Quadratic extensions
11R23Iwasawa theory
11D09Quadratic and bilinear diophantine equations
Full Text: DOI
[1] Ankeny, N. C.; Chowla, S.; Hasse, H.: On the class number of the maximal real subfield of a cyclotomic field. J. reine angew. Math. 217, 217-220 (1965) · Zbl 0128.03501
[2] Borevich, Z. I.; Shafarevich, I. R.: Number theory. (1966) · Zbl 0145.04902
[3] Cowles, M. J.: On the divisibility of the class number of imaginary quadratic fields. J. number theory 12, 113-115 (1980) · Zbl 0427.12001
[4] Degert, G.: Über die bestimmung der grundeinheit gewisser reel-quadratischer zahlokörper. Abh. math. Sem. univ. Hamburg 22, 92-97 (1958) · Zbl 0079.05803
[5] Garbanati, D.: An algorithm for finding an algebraic number whose norm is a given rational number. J. reine angew. Math. 316, 1-13 (1980) · Zbl 0421.12002
[6] Gross, B. H.; Rohrlich, D. E.: Some results on the Mordell-Weil group of the Jacobian of the Fermat curve. Invent. math. 44, 201-224 (1978) · Zbl 0369.14011
[7] Janusz, G. J.: Algebraic number fields. (1973) · Zbl 0307.12001
[8] Lang, S. D.: Note on the class-number of the maximal real subfield of a cyclotomic field. J. reine angew math. 290, 70-72 (1977) · Zbl 0346.12003
[9] Mollin, R. A.: On the cyclotomic polynomial. J. number theory 17, 165-175 (1983) · Zbl 0512.10006
[10] Mollin, R. A.: Class numbers and a generalized Fermat theorem. J. number theory 16, 420-429 (1983) · Zbl 0515.12004
[11] Nagell, T.: Introduction to number theory. (1964) · Zbl 0042.26702
[12] Takeuchi, H.: On the class-number of the maximal real subfield of a cyclotomic field. Canad. J. Math. 33, No. No. 1, 55-58 (1981) · Zbl 0482.12004
[13] Yamaguchi, I.: On the class-number of the maximal real subfield of a cyclotomic field. J. reine angew. Math. 272, 217-220 (1975) · Zbl 0313.12003
[14] Yokoi, H.: On the Diophantine equation $x2-py2={\pm}$4q and the class number of real subfields of a cyclotomic field. Nagoya math. J. 91, 151-161 (1983) · Zbl 0506.10012