Let D be a Euclidean domain with Euclidean valuation v If u

Let D be a Euclidean domain with Euclidean valuation v If u is a unit in D, show that v(u) = v(1).

Solution

We know that a Euclidean domain is defined by D and the valuation is denoted by v respectively. There are two conditions that we might have to concern ourselves with regarding if for some a,b nonzero elements in D then v(a) is less than or equal to v(ab). Also, if a,b are in D and b is not 0 then there must be some q,r in D that satisfy the Euclidean algorithm a=bq+r and where r=0 or v(r)<v(b).

Suppose R is a Euclidean domain and let IR be the set in the question. Then either I={0}=(0) or we can take a a0 in I with v(a) minimal. Then for any bI, we can write b=qa+r with r=0 or v(r)<v(a). But r=bqaI and so by minimality of v(a), r=0; thus ab and I=(a). hence v(u)=v(I)