Describe the maximal ideals in the ring of Gaussian Integers $\Bbb Z[i]$.

Question

Describe the maximal ideals in the ring of Gaussian Integers $\Bbb Z[i]$.

So first of all my question would be - Is it possible to write any ideal of $\Bbb Z[i]$ (which happens to be a PID) as $\langle a\rangle$ or does it have to be $\langle a + bi\rangle$ or are they the same thing?

i.e Can ANY Ideal in $Z[i]$ be written as $\langle a\rangle$ ?

Further, I am not able to continue. Please help!

Some possible references (I looked through these but to no avail):

1. Maximal ideals in the ring of Gaussian integers

2. How to find all the maximal ideals of $\mathbb Z_n?$

3. Ideals in the ring of Gaussian integers

Answer 1

Since $\mathbf Z[i]$ is a P.I.D. – actually a Euclidean domain with the norm: $N(a+bi)=a^2+b^2$ as a Euclidean function, maximal ideals are generated by irreducible elements.

Also $a+bi$ is irreducible if $N(a+bi)$ is prime, because the norm is multiplicative, i.e. $$N\bigl((a+bi)(c+di)\bigr)=N(a+bi)N(c+di).$$

Answer 2

No; not every ideal of $\Bbb{Z}[i]$ is of the form $\langle a\rangle$ for some integer $a$. A simple example is the ideal $\langle 1+i\rangle$. It contains the integer $2=(1+i)(1-i)$, so if it is generated by an integer, then it must be generated by a divisor of $2$. But then it should equal either $\langle1\rangle=\Bbb{Z}[i]$ or $\langle2\rangle$, which it doesn't.