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

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 $$\langle a\rangle$$ ?

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

• Use the definition of PID. and an Ideal Commented Jul 14, 2019 at 11:41
• @Kumar Could not conclude anything substantial from the above link.
– AJ_
Commented Jul 14, 2019 at 11:54
• What do you denote $a$ in $\langle a\rangle$? An ordinary integer or a Gauß integer? Commented Jul 14, 2019 at 11:58
• I think if you look at the question and answer of the link, you would be able to get the answer. Anyways. math.stackexchange.com/questions/764437/… This link will help you to understand more if you understand polynomial rings. I would also recommend you to have a look at Dummit and Foote, Abstract Algebra. Happy idealizing. :P Commented Jul 14, 2019 at 12:04
• The maximal ideals of $\Bbb{Z}[i]$ are those generated by the primes of this ring. Whether a rational (=usual) prime $p$ remains a prime in $\Bbb{Z}[i]$ depends on its residue class modulo $4$. If $p\not\equiv-1\pmod4$, then it factors further. Commented Jul 14, 2019 at 17:15

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. $$N\bigl((a+bi)(c+di)\bigr)=N(a+bi)N(c+di).$$
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.