nLab Dedekind-Hasse norm

Contents

Contents

Idea

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.

Definition and characterization of pids

Definition

Let RR be an integral domain. A Dededekind-Hasse norm is a function v:Rv \colon R \rightarrow \mathbb{N} such that:

  • v(s)=0s=0v(s)=0 \Leftrightarrow s=0

  • aR\forall a \in R, (b0)R\forall (b \neq 0) \in R:

    • b|ab|a, or

    • p,q,rR\exists p,q,r \in R such that pa=bq+rpa = bq + r and 0<v(r)<v(b)0 \lt v(r) \lt v(b)

Proposition

Let RR be an integral domain. Then it is a principal ideal domain iff it possesses a Dedekind-Hasse norm.

Proof

Suppose that RR possesses a Dedekind-Hasse norm. Let II be a non-zero ideal. Let bb be a non-zero element of II of minimal norm. We know that (b)I(b) \subseteq I. Let aa be an element of II. Suppose that bb doesn’t divide aa. Then, there exists p,q,rRp,q,r \in R such that pa=bq+rpa = bq+r and 0<v(r)<v(b)0 \lt v(r) \lt v(b). Thus, r=pabqIr=pa-bq \in I, r0r \neq 0 and v(r)<v(b)v(r) \lt v(b), absurd! Therefore b|ab|a and a(b)a \in (b). Thus, I(b)I \subseteq (b) and I=(b)I = (b). We have proved that RR is a pid.

Suppose that RR is a pid. Thus, it is a UFD. Put v(s)=0v(s)=0 if s=0s=0, and v(s)v(s) equal to 2 n2^{n} where nn is the number of irreducible elements in the factorization of nn, if s0s \neq 0. Let aRa \in R and (b0)R(b \neq 0) \in R. Suppose that bb doesn’t divide aa. We know that (a,b)=(r)(a,b) = (r) and thus there exists p,q,rRp,q,r \in R such that pa+bq=rpa+bq = r. rr divides bb but bb doesn’t divide rr because it would imply that bb divides aa. Thus, there is strictly less irreducible elements in the factorization of rr than in the one of bb and v(r)<v(b)v(r) \lt v(b). Moreover r0r \neq 0 because (a,b)=(r)(a,b) = (r) and b0b \neq 0. Thus 0<v(r)<v(b)0 \lt v(r) \lt v(b). We have proved that vv is a Dedekind-Hasse norm.

References

Named after Helmut Hasse.

Last revised on August 13, 2022 at 15:49:08. See the history of this page for a list of all contributions to it.