Abstract
LetD be a Dedekind domain. It is well known thatD is then an atomic integral domain (that is to say, a domain in which each nonzero nonunit has a factorization as a product of irreducible elements). We study factorization properties of elements in Dedekind domains with finite class group. IfD has the property that any factorization of an elementα into irreducibles has the same length, thenD is called a half factorial domain (HFD, see [41]). IfD has the property that any factorization of an elementα into irreducibles has the same length modulor (for somer>1), thenD is called a congruence half factorial domain of orderr. In Section I we consider some general factorization properties of atomic integral domains as well as the interrelationship of the HFD and CHFD property in the Dedekind setting. In Section II we extend many of the results of [41], [42] and [36] concerning HFDs when the class group ofD is cyclic. Finally, in Section III we consider the CHFD property in detail and determine some basic properties of Dedekind CHFDs. IfG is any Abelian group andS any subset ofG−[0], then {G, S} is called a realizable pair if there exists a Dedekind domainD with class groupG such thatS is the set of nonprincipal classes ofG which contain prime ideals. We prove that for a finite abelian groupG there exists a realizable pair {G, S} such that any Dedekind domain associated to {G, S} is CHFD for somer>1 but not HFD if and only ifG is not isomorphic toZ 2,Z 2,Z 2 ⊕Z 2, orZ 3 ⊕Z 3.
Similar content being viewed by others
References
L. Carlitz,A characterization of algebraic number fields with class number two, Proc. Am. Math. Soc.11 (1960), 391–392.
S. Chapman and W.W. Smith,On a characterization of algebraic number fields with class number less than three, J. Algebra, to appear.
L. Claborn,Every abelian group is a class group, Pacific J. Math.18 (1966), 219–222.
L. Claborn,Specific relations in the ideal group, Michigan Math. J.15 (1968), 249–255.
P. M. Cohn,Bezout rings and their subrings, Proc. Camb. Phil. Soc.64 (1968), 251–264.
P. M. Cohn,Unique factorization domains, Am. Math. Monthly80 (1973), 1–17.
A. Czogala,Arithmetic characterization of algebraic number fields with small class number, Math. Z.176 (1981), 247–253.
F. DiFranco and F. Pace,Arithmetical characterization of rings of algebraic integers with class number three and four, Boll. Un. Mat. Ital. D(6)4 (1985), 63–69.
R.M. Fossum,The Divisor Class Group of a Krull Domain, Springer-Verlag, Berlin, 1973.
A. Geroldinger,Über nicht-eindeutige Zerlegungen in irreduzible Elemente, Math. Z.197 (1988), 505–529.
R. Gilmer,Multiplicative Ideal Theory, Marcel-Dekker, New York, 1972.
A. Grams,Atomic rings and the ascending chain condition for principal ideals, Proc. Camb. Phil. Soc.75 (1974), 321–329.
A. Grams,The distribution of prime ideals of a Dedekind domain, Bull. Aust. Math. Soc.11 (1974), 429–441.
E. Hecke,Über die L-Funktionen und den Dirichletschen Primzahlsatz für einen beliebigen Zahlkörper, Nachr. Akad. Wiss Gottigen. Math.-Phys. Kl. IIa (1917).
J. Kaczorowski,A pure arithmetical characterization for certain fields with a given class group, Colloq. Math.45 (1981), 327–330.
U. Krause,A characterization of algebraic number fields with cyclic class group of prime power order, Math. Z.186 (1984), 143–148.
D. Michel and J. Steffan,Répartition des idéaux premiers parmi les classes d’idéaux dans un anneau de Dedekind et équidécomposition, J. Algebra98 (1986), 82–94.
W. Narkiewicz,A note on factorizations in quadratic fields, Acta Arith.15 (1968), 19–22.
W. Narkiewicz,A note on numbers with good factorization properties, Colloq. Math.17 (1973), 275–276.
W. Narkiewicz,Class number and factorization in quadratic number fields, Colloq. Math.17 (1967), 167–190.
W. Narkiewicz,Elementary and Analytic Theory of Algebraic Numbers, PWN-Polish Scientific Publishers, Warsaw, 1974.
W. Narkiewicz,Finite Abelian groups and factorization problems, Colloq. Math.42 (1979), 319–330.
W. Narkiewicz,Numbers with unique factorization in an algebraic number field, Acta Arith.21 (1972), 313–322.
W. Narkiewicz,On algebraic number fields with non-unique factorization, Colloq. Math.12 (1964), 59–68.
W. Narkiewicz,On algebraic numbers fields with non-unique factorization, II, Colloq. Math.15 (1966), 49–58.
W. Narkiewicz,Some unsolved problems, Bull. Soc. Math. France25 (1971), 159–164.
W. Narkiewicz and J. Sliwa,Finite Abelian groups and factorization problems II, Colloq. Math.46 (1982), 115–122.
J. E. Olsen,A combinatorial problem in finite abelian groups. I, J. Number Theory1 (1969), 8–10.
J. E. Olsen,A combinatorial problem in finite abelian groups. II, J. Number Theory1 (1969), 195–199.
D. Rush,An arithmetic characterization of algebraic number fields with a given class group, Math. Proc. Camb. Phil. Soc.94 (1983), 23–28.
L. Salce and P. Zanardo,Arithmetical characterization of rings of algebraic integers with cyclic ideal class group, Boll. Un. Mat. Ital. D(6)1 (1982), 117–122.
P. Samuel,On unique factorization domains, Illinois J. Math.5 (1961), 1–17.
P. Samuel,Sur les anneaux factoriels, Bull. Soc. Math. France89 (1961), 155–173.
P. Samuel,Unique factorization, Am. Math. Monthly75 (1968), 945–952.
L. Skula,Divisorentheorie einer Halbgruppe, Math. Z.114 (1970), 113–120.
L. Skula,On c-semigroups, Acta Arith.31 (1976), 247–257.
J. Sliwa,Factorizations of distinct lengths in algebraic number fields, Colloq. Math.31 (1976), 399–417.
J. Sliwa,Remarks on factorizations in algebraic number fields, Colloq. Math.46 (1982), 123–130.
J. Steffan,Longueurs des décompositions en produits d’éléments irréductibles dans un anneau de Dedekind, J. Algebra102 (1986), 229–236.
A. Zaks,Atomic rings without a.c.c. on principal ideals, J. Algebra74 (1982), 223–231.
A. Zaks,Half factorial domains, Isr. J. Math.37 (1980), 281–302.
A. Zaks,Half factorial domains, Bull. Am. Math. Soc.82 (1976), 721–723.
Author information
Authors and Affiliations
Additional information
The first author received support under the John M. Bennett Fellowship at Trinity University and also gratefully acknowledges the support of The University of North Carolina at Chapel Hill.
Rights and permissions
About this article
Cite this article
Chapman, S.T., Smith, W.W. Factorization in dedekind domains with finite class group. Israel J. Math. 71, 65–95 (1990). https://doi.org/10.1007/BF02807251
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02807251