Is $\Bbb Z[i]$ a Euclidean ring?

Question

Is $\Bbb Z[i]$ a Euclidean ring?

If not, what would be the simplest way of seeing that $\Bbb Z[i]$ is a PID?

Answer 1

Example of division in $\Bbb Z[i]$:

$$\frac{11+4i}{2+5i}=\frac{42}{29}-\frac{47}{29}i$$

The nearest Gaussian integer is $2-2i$, so this is the Eculidean quotient:

$$11+4i=(2-2i)(2+5i)+(-3-2i)$$

The remainder is $(-3-2i)$.

Of course, this does not prove that $\Bbb Z[i]$ is an Euclidean domain, but you can see how the division is. You can try to prove that this is a valid Euclidean division. The norm of a Gaussian integer for this division is the square of its modulus.