Composition Algebras over Rings of Fractions Revisited

Letkbe a field of characteristic not two, letfh(x0, x1) k[x0, x1] be an irreducible homogeneous polynomial and denote the ring of elements of degree zero in the homogeneous localizationk[x0, x1]fhbyk[x0, x1](fh). For deg fh = 3 it is proved that the composition algebras overk[x0, x1](fh)not containing zero divisors are defined overkand that there is at most one (split) composition algebra not defined overk. For deg fh = 4 all composition algebras overk[x0, x1](fh)are enumerated and partly classified.