This is lemma 5 in section 12.2 from Ireland and Rosen's "A Classical Introduction to Modern Number Theory". $D$ is the ring of algebraic integers of an algebraic number field $F$, and $N(x)$ denotes the norm of $x$ with respect to $F/\mathbb Q$.
Why do we allow $M$ to be dependent on $\gamma$, if $M$ isn't dependent on $\alpha, \beta$?