## The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain

## Problem 503

Prove that the ring of integers

\[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\]
of the field $\Q(\sqrt{2})$ is a Euclidean Domain.

Prove that the ring of integers

\[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\]
of the field $\Q(\sqrt{2})$ is a Euclidean Domain.

For each positive integer $n$, prove that the polynomial

\[(x-1)(x-2)\cdots (x-n)-1\]
is irreducible over the ring of integers $\Z$.

Let $\Z$ be the ring of integers and let $R$ be a ring with unity.

Determine all the ring homomorphisms from $\Z$ to $R$.