mathmari said:
We have the following:
So, the maximal ideals are $(2)$ and $(3)$, right? (Wondering)
You should also have a line going between $(2)$ and $(6)$ since $(6) \subset (2)$, as well.
All finite fields have order $p^n$. So, there cannot be a finite field of order $6$.
Indeed. Nor is there a field or order 12.
We have that $\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}_n$.
But now we have $\mathbb{Z}_{12}$ instead of $\mathbb{Z}$.
Suppose that $n=2$, then $2\mathbb{Z}_{12}=\{0, 2, 4, 6, 8, 10\}$. Does $\mathbb{Z}_{12}/2\mathbb{Z}_{12}$ contain then all the elements of $\mathbb{Z}_{12}$ modulo the even ones? (Wondering)
Let's look at an example: $\Bbb Z_{12}/3\Bbb Z_{12}$. We need to look at the cosets of:
$H = (3) = \{0,3,6,9\}$ ($H$ is clearly our first coset). Since $1 \not\in H$, we have $1+H$ is another coset:
$1 + H = \{1,4,7,10\}$. In the same manner, we have a third coset:
$2 + H = \{2,5,8,11\}$. We have now completely partitioned $\Bbb Z_{12}$, so that's all.
Note that for $a \in 1 + H$, and $b \in 2 +H$, we have $a + b \in H$, for example:
$7 + 5 = 0$, and $4 + 11 = 3$.
This is purely an informal observation, but I believe (because I have faith in you!) you can prove:
$\Bbb Z_{12}/3\Bbb Z_{12} \cong \Bbb Z_3$ (as RINGS!).
Now $\Bbb Z_3$ is indeed a field, so $(3)$ is a prime ideal.
Now in this case, we could have just argued by order:
$|\Bbb Z_{12}/3\Bbb Z_{12}| = \dfrac{|\Bbb Z_{12}|}{|3\Bbb Z_{12}|} = \dfrac{12}{4} = 3$, and any group of order 3 is isomorphic (as a group) to $\Bbb Z_3$, and it is not hard to see that corresponding multiplication of multiplication (mod $3$) in $\Bbb Z_3$ is "the same" as the coset multiplication in $\Bbb Z_{12}/3\Bbb Z_{12}$, that is:
$k + H \mapsto k$ (mod $3$) is a RING-HOMOMORPHISM (and thus an isomorphism since it's bijective).
The case of $(4)$ is an exception, one cannot argue purely by order, because the resulting quotient has order 4, and there do exist fields of order 4. But $(4)$ is not a prime ideal, do you see why?
$(0)$ is another interesting case, one that people often over-look. It's not prime...but why?
Finally, why is $(1)$ (another "edge case") not a prime ideal?