# How to prove that an element of the minimum norm in Euclidean Domain is a unit

The best way to prove this is by taking the definition of a Euclidean Domain. We need to take into account that the minimum norm is nonzero.## Prove that an element of minimum norm in Euclidean Domain is a unit

**Proof:**let R be an Euclidean Domain and let N(x) be the be the nonzero minimum norm of x. By definition of the Euclidean Domain, we have the following:

\begin{equation*} a = qx + r, \quad \text{where } r = 0 \text{ or } N(r) < N(x). \end{equation*}