Let $k$ be any field and let $A = k[X,Y,Z]/(X^2 - Y^3 - 1, XZ - 1)$.
How can I find $\alpha, \beta \in k$ such that $A$ is integral over $B = k[X + \alpha Y + \beta Z]$?
For these values of $\alpha$ and $\beta$, how can I find concrete generators for $A$ as a $B$-module?
Denote by $x, y, z$ the respective classes of $X, Y, Z$ in $A$. Then $A$ is finite over $k[x,z]$ because $y^3=x^2-1$. As $xz=1$, we have $k[x,z]$ finite over $k[x+z]$ because $x, z$ are both solutions of $T^2-(x+z)T+1$. So $A$ is finite hence integral over $B:=k[X+Z]$. As
$$A=k[x,z]+k[x,z]y+k[x,z]y^2, \quad k[x,z]=k[x+z]+k[x+z]x+k[x+z]z,$$ it is easy to find a set of generators of $A$ as $B$-module.