Consider a field $F$ of characteristic zero. Let $L=F[\alpha]$ be an extension of degree $d.$ We call an element $$ x=x_0 + x_1 \alpha +\ldots+ x_{d-1}\alpha^{d-1}\in L $$ short if $x_{d-1}=0.$ Under which conditions on $\alpha$ every element in $L^\times$ is a product of short elements?
It is easy to see that every element of $L$ is a quotient of two short elements for $d\geq 3$.