Abstract Algebra: Theory and Applications · Problem 18.4.15

Lattices and Boolean Algebras — Thomas Judson

Mathematics — Abstract Algebra · euclidean-valuation-associates

Problem

Let DD be a Euclidean domain with Euclidean valuation ν\nu. If aa and bb are associates in DD, prove that ν(a)=ν(b)\nu(a) = \nu(b). Hint. Let a=uba = ub with uu a unit. Then ν(b)≤ν(ub)≤ν(a)\nu(b) \leq \nu(ub) \leq \nu(a). Similarly, ν(a)≤ν(b)\nu(a) \leq \nu(b).

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