symmetric monoidal (∞,1)-category of spectra
The notion of a Dedekind-Hasse norm is a generalization of the notion of the degree function in a Euclidean domain. It makes clear the links between a principal ideal domain and an Euclidean domain: a pid is like an Euclidean domain but with a weaker notion of Euclidean division.
Let be an integral domain. A Dededekind-Hasse norm is a function such that:
, :
, or
such that and
Let be an integral domain. Then it is a principal ideal domain iff it possesses a Dedekind-Hasse norm.
Suppose that possesses a Dedekind-Hasse norm. Let be a non-zero ideal. Let be a non-zero element of of minimal norm. We know that . Let be an element of . Suppose that doesn’t divide . Then, there exists such that and . Thus, , and , absurd! Therefore and . Thus, and . We have proved that is a pid.
Suppose that is a pid. Thus, it is a UFD. Put if , and equal to where is the number of irreducible elements in the factorization of , if . Let and . Suppose that doesn’t divide . We know that and thus there exists such that . divides but doesn’t divide because it would imply that divides . Thus, there is strictly less irreducible elements in the factorization of than in the one of and . Moreover because and . Thus . We have proved that is a Dedekind-Hasse norm.
Named after Helmut Hasse.
Last revised on August 13, 2022 at 19:49:08. See the history of this page for a list of all contributions to it.