Elasticity of Factorization in Number Fields
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Introduction De
nition A Dedekind domain O is an integral domain, which is Noetherian, integrally closed, and of dimension 1. De
nition Let O be a ring and K its quotient
eld. A fractional ideal of O in K is an O-module a contained in K such that there exist an element d 2 Onf0g for which da O. Theorem 1. If O is a Dedekind domain, then every ideal of O can be uniquely factored into prime ideals, and the non-zero fractional ideals form a group under multiplication, I (O). De
nition Let O be a Dedekind domain and K its quotient
eld. Let ' : K ! I (O) be de
ned by ' (u) = uO. De
ne P (O) = Im' to be the group of principal fractional ideals of O. The ideal class group of O is de
ned to be Cl (O) = I (O) =P (O) and the class number of O is hO = jCl (O)j. When O is clear from context, we shall omit reference to O and simply use Cl; I; P; and h. In particular, we have the short exact sequence, 1 ! P ! I ! Cl ! 1:
[1] Robert J. Valenza,et al. Elasticity of factorization in number fields , 1990 .
[2] L. Carlitz,et al. A CHARACTERIZATION OF ALGEBRAIC NUMBER FIELDS WITH CLASS NUMBER TWO , 1960 .