ABSTRACT
The term semifield will refer to a not necessarily associative division ring. Semi-fields are used in the study of projective planes. J. Petit presented a construction for semifields based on results contained in O. Ore’s papers on rings of noncommutative polynomials. This chapter reviews Petit’s construction, giving a proof of his existence criteria (no proof is given in his papers), and gives explicit construction for semifields. The projective planes of Lens-Barlotti type V.l are those translation planes (with translation line uv) which are also dual translation planes (with translation point U). These are precisely the planes which can be coordinatized using semifields. Petit’s method can be used to construct semifields three dimensional over their left nuclei that are not algebras three dimensional over their nuclei.
