(This question is a dupplicate from If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.)
I'm trying to solve this problem with my current knowledge of algebraic integers:
- $x$ is an algebraic integer if it is a root of a monic polynomial in $\mathbb Z[x]$
- $x$ is an algebraic integer iff $xW\subset W$ for some finitely generated free $\mathbb Z$-module $W$ (i.e. $W$ has a finite basis).
My first thought was to try Hurkyl's answer and compute $$ \prod_i\prod_j x^2+\alpha_ix+\beta_j $$ where $\alpha_i$ and $\beta_j$ are all the (complex) roots of the minimal polynomials of $\alpha$ and $\beta$ but I couldn't show that it has integer coefficients. (Although I found that $\prod_j x^2+\alpha x+\beta_j=p_\beta(-x^2-\alpha x)$ where $p_\beta$ is the minimal polynomial of $\beta$.)
Today my teacher gave me as hint to take a smartly chosen $\mathbb Z$-module and use the second definition given above. However I don't see which one...
Could someone give me a hint?