About number fields with Pólya group of order ≤2

Abstract

Carlitz characterized the number fields K with class number ≤2 by the equality of the lengths of all the factorizations of every integer of K into irreducible elements. Analogously, we study the links between the order of the Pólya group Po(K) of a number field K and the factorizations into irreducible elements of some rational numbers. Our main results concern quadratic fields where we prove some equivalences between, on the one hand, |Po(K)| = 1 and uniqueness of factorizations, on the other hand, |Po(K)| = 2 and uniqueness of lengths of factorizations. We also show how analogous results may be formulated in the case of function fields.