Composition algebras over rings of fractions revisited (Q1268092)

For \(k\) a field of characteristic not two, let \(f_h (x_0,x_1)\in k[x_0,x_1]\) denote an irreducible homogeneous polynomial and the ring of elements of degree zero in the homogeneous localization \(k[x_0,x_1]_{f_h}\) is denoted by \(k[x_0,x_1]_{(f_h)}\). For \(\text{deg}f_h=3\) the author proves that the composition algebras over \(k[x_0,x_1]_{(f_h)}\) not containing zero divisors are defined over \(k\) and that there is at most one (split) composition algebra not defined over \(k\). For \(\text{deg}f_h=4\) all composition algebras over \(k[x_0,x_1]_{(f_h)}\) are enumerated and partly classified.