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*}
In a Euclidean domain, it must hold for any
a and
x, so we can take
a = 1. Now if
r = 0, then we have that
x is a unit. If
N(r) <
N(x), then
N(r) is smaller than the nonzero minimum norm
N(x). This implies that
N(r) must be zero. Therefore,
x is a unit.
Conclusion
For this type of questions, it is important to use definitions.
