Elasticity of factorization in number fields. (English) Zbl 0721.11043

Valenza, R. J.

J. Number Theory 36, No. 2, 212-218 (1990).

For a Noetherian domain \(R\) (which is not a field) the author defines the elasticity of \(R\) to be \(\sup(X)\) where \(X\) is the subset of \(\mathbb Q\) defined by \(X=\{m/n : \) there are irreducibles \(\pi_ 1,\pi_ 2,\ldots,\pi_ m\), \(\tau_ 1,\tau_ 2,\ldots,\tau_ n\in R\) with \(\pi_ 1\pi_ 2\cdots\pi_ m=\tau_ 1\tau_ 2\cdots\tau_ n\}\); whence, \(\rho(R)\) measures the failure of \(R\) to be a unique factorization domain. This is related to the well-known Carlitz criterion which says that \(\rho(A)=1\) if and only if \(h\leq 2\) where \(h\) is the class number (or order of the class group \(C_ K)\) of the ring of integers \(A\) of an algebraic number field \(K\). The author proves three results among which is that \(\rho(A)\) is bounded from above by \(h/2\) whenever \(C_ K\neq 1\). The author defines a new group invariant called sequential depth to prove this.

Reviewer: Richard A. Mollin (Calgary)

MSC:

11R27	Units and factorization
11R29	Class numbers, class groups, discriminants
13F15	Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)

Keywords:

Noetherian domain; unique factorization domain; Carlitz criterion; class number; sequential depth

References:

[1]	Carlitz, L., A characterization of algebraic number fields with class number two, (Proc. Amer. Math. Soc., 11 (1960)), 391-392 · Zbl 0202.33101
[2]	Janusz, G. J., (Algebraic Number Fields (1973), Academic Press: Academic Press New York) · Zbl 0307.12001
[3]	Kasube, H. E., Unique and almost unique factorization, (Natanson, M. B., Proceeding of the Southern Illinois Number Theory Conference. Proceeding of the Southern Illinois Number Theory Conference, Carbondale, March 30 and 31, 1979. Proceeding of the Southern Illinois Number Theory Conference. Proceeding of the Southern Illinois Number Theory Conference, Carbondale, March 30 and 31, 1979, Lecture Notes in Mathematics, Vol. 751 (1980), Springer-Verlag: Springer-Verlag Berlin/New York), 200-205 · Zbl 0421.12003
[4]	Masley, J. M., Where are number fields with small class number?, (Natanson, M. B., Proceeding of the Southern Illinois Number Theory Conference. Proceeding of the Southern Illinois Number Theory Conference, Carbondale, March 30 and 31, 1979. Proceeding of the Southern Illinois Number Theory Conference. Proceeding of the Southern Illinois Number Theory Conference, Carbondale, March 30 and 31, 1979, Lecture Notes in Mathematics, Vol. 751 (1980), Springer-Verlag: Springer-Verlag Berlin/New York), 221-242 · Zbl 0421.12005