An algebraic integer is a complex number that is a root of a monic polynomial with coefficients in $\mathbb{Z}$.
Let $\alpha$ and $\beta$ be algebraic integers. Then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.
What we have so far is that $2x+\alpha$ is an algebraic integer but since the set of algebraic integers is a ring I can't divide by 2 so I can't have an algebraic integer and also the polynomial wouldn't be monic.
I appreciate the help.
Best Answer
If I well remember, there's a characterization of integral elements over a ring R (in our case $R=\mathbb{Z}$), in particular $x$ is integral over $R$ if and only if it is contained in a ring $ C$ such that $R \subset C$ and $C$ is a finite algebra over $R$.
In our case a rooth of $x^2+\alpha x+\beta$ is integral over $\mathbb{Z}[\alpha,\beta]$ and so there exist a ring $C$ with the property above.
Now $\mathbb{Z}[\alpha,\beta]$ is a finite $\mathbb{Z}$-module, because $\alpha$ and $\beta$ are algebraic integers. Then $C$ is a finite $\mathbb{Z}$-module and $x$ is an algebraic integer.