nt.number theory – Products of short elements in a field

Consider a field $F$ of characteristic zero. Let $L=F[alpha]$ be an extension of degree $d.$ We call an element
$$
x=a_0 + a_1 alpha +ldots+ a_{d-1}alpha^{d-1}in L
$$

short if $a_{d-1}=0.$
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 $dgeq 3$.