ELASTICITY OF FACTORIZATION IN NUMBER-FIELDS

Let R be a noetherian domain which is not a field. Define ρ{variant}(R), the elasticity of R, to be sup(X), where X is the subset of Q defined by X={m/n: there exist irreducibles π1, π1, ..., πm, t1, t2, ..., tn ∈ R with π1π2⋯ πm = t1π2⋯tn} Thus ρ{variant}(R) in part measures the failure of R to be a unique factorization domain. Let A denote the integers of K, a number field with class group C and class number h. By elementary number theory and a theorem of L. Carlitz (Proc. Amer. Math. Soc. 11 (1960), 391-392) one knows that ρ{variant}(A) = 1 if, and only if, h ≤ 2. We show here that 1. (i) ρ{variant}(A) is bounded from above by h 2, provided that C is nontrivial. 2. (ii) ρ{variant}(A) is bounded from below by an expression which depends only on the elementary divisor decomposition of C and which is greater than or equal to half the exponent of C; in particular, this agrees with the upper bound given in (i) in the special case that C is cyclic. 3. (iii) ρ{variant}(A) → ∞ as h → ∞. The actual bound in (i) is somewhat sharper and involves a newly defined group invariant called sequential depth, which is fundamental to our discussion. © 1990.