ABSTRACT
Abstract Little is known of the structure of finite division rings. The major results are the classification of finite division rings quadratic over a weak nucleus by Knuth [15] and amplified in the papers by Cohen and Ganley [5] and Ganley [11], the determination of sufficient conditions for the existence of subrings by Zemmer [23] and the proof that all flexible finite division rings of characteristic not 2 are commutative by Oehmke [17] . The traditional method of classifying finite division rings is to give the discoverer’s name, i.e., Dickson commutative division rings [10] or Sandler division rings [20]. Our examples highlight some of the differences between finite fields and finite division rings and demonstrate that isotopic division rings need not have isomorphic automorphism groups nor isomorhpic nuclei. One example is that of a division ring of order 81 that contains no subring of order 9 or 27. We will examine structure and automorphism groups in preparation to establishing a coherent theory of finite division rings.
