I am trying to prove that in Euclidean domain D with Euclidean function d, u in D is a unit if and only if d(u)=d(1).
Suppose u is a unit, then there exist v in D such that uv=1, this implies u\1 so d(u)<=d(1), but obviously 1 divides u so d(1)<=d(u). Hence, d(u)=d(1).
Conversely, suppose d(u)=d(1), since u is not zero, there exist q and r in D such that 1=uq+r with r=0 or d(r)< d(u).
If r=0 then u is a unit. Else d(r)< d(u) =d(1), this implies d(r)< d(1). I stop here, because I failed to argue that r must be zero.
Can anyone help me? Thanks.