I'm really stuck in this exercise. Hope you can help me somehow. Any advice would be very nice.
Let $$\overline{\mathbb{Z}}:=\{z \in \mathbb{C}\mid f(z)=0 \text{ for a monic polynomial } f\in \mathbb{Z}[X]\}\subset\overline{\mathbb{Q}}\subset\mathbb{C}.$$
Show that:
a) $\overline{\mathbb{Z}}$ is a ring with no irreducible elements.
b) Every prime ideal $I \ne \{ 0 \}$ in $\overline{\mathbb{Z}}$ is maximal.
c) Prove whether $ \frac{-1+\sqrt{3}}{2}$ and $ \frac{-1+\sqrt{-3}}{2}$ are in $\overline{\mathbb{Z}}$.
At first, for a) "Showing that $\overline{\mathbb{Z}}$ is a ring" I would start with, showing that for $\alpha, \beta \in \overline{\mathbb{Z}}$, $-\alpha, \alpha +\beta$ and $\alpha \beta$ is in $\overline{\mathbb{Z}}$.
I have already proved it for the easy first case : $-\alpha$.
Now for $\alpha + \beta$: Suppose that $\alpha$ is the root of a polynomial, named $f$ and $\beta$ is the root of a polynomial, named $g$. Let $\alpha_1,…, \alpha_n$ be the set of the entire roots from $f$ and $\beta_1,…,\beta_m$ the roots from $g$. Now I would consider the polynomial:
$$h(X)=\prod \limits_{i=1}^{n}\prod \limits_{j=1}^{m} (X-(\alpha_i+\beta_j)).$$
At that point I don't know how to prove that the coefficients of $h$ are in $\mathbb{Z}$.
For $\alpha\beta$ it should work with $$\prod \limits_{i=1}^{n}\prod \limits_{j=1}^{m} (X-\alpha_i \beta_j),$$ but here I have a similar problem.
For the rest of a) and particularly in b) I have no clue.
I also tried to find some polynomials in c) but always failed. What comes to in my mind is that if we showed a, then we can use the fact that $\overline{\mathbb{Z}}$ is a ring and maybe first proof that $ \frac{-1}{2}$ is in $\overline{\mathbb{Z}}$ and then $\frac{\sqrt{3}}{2}$, but it's not so easy for me because those polynomials should be monic…
As you see I have many problems. Thank you very much, even if you can help a little.
Best Answer
Hint for a and c:
Prove that $\alpha\in \overline{\mathbf Z}\;$ if and only the ring $\mathbf Z[\alpha]$ is a finitely generated $\mathbf Z$-module.
No irreducible elements: just show the square root of an algebraic integer is an algebraic integer.
b) Show that any element $\alpha\notin \mathfrak p$ (a non-zero prime ideal of $\overline{\mathbf Z}$) is a unit modulo $\mathfrak p$. For that, show that $\mathfrak p\cap\mathbf Z[\alpha] $ is a maximal ideal of $\mathbf Z[\alpha] $.
Sub-hint: Set $\mathfrak p\cap\mathbf Z=p\mathbf Z$ and show the quotient $\mathbf Z[\alpha]/\mathfrak p\cap\mathbf Z[\alpha]$ is a finite-dimensional $\mathbf Z/p\mathbf Z$-vector space.
c) Compute the minimal polynomials of each element.