On K-derived quartics

Let K be a number field. A K  -derived polynomial f(x)∈K[x]f(x)∈K[x] is a polynomial that factors into linear factors over K, as do all of its derivatives. Such a polynomial is said to be proper   if its roots are distinct. An unresolved question in the literature is whether or not there exists a proper QQ-derived polynomial of degree 4. Some examples are known of proper K-derived quartics for a quadratic number field K  , though other than Q(3), these fields have quite large discriminant. (The second known field is Q(3441).) The current paper describes a search for quadratic fields K over which there exist proper K  -derived quartics. The search finds examples for K=Q(D) with D=…,−95,−41,−39,−19,21,31,89,…D=…,−95,−41,−39,−19,21,31,89,… .