An algebraic system which satisfies every condition of a skewfield but only one (say: the left) and not necessarily both of the distributive laws is called a nearfield. The existence of proper nearfields (which are not skewfields) was first noted by L. E. Dickson. In the expository paper [3] T.Y. Lam works out the structure of the smallest proper nearfield, the multiplicative group of which is a quaternion group of order 8. By transferring the addition of the nearfield to Q 8 0} = 0, ±1, ±i, ±j, ±k} he makes the connection to Hamilton's quaternions. Starting-point of the present paper was the insight that this addition in Q 80} can be described in a natural way by congruences modulo a prime quaternion in the Hurwitz order mathcalO} of integral quaternions. First we consider nearfields F and study the properties of the subring R of End(F, +) generated by all left multiplications λa : x mapsto ax (x in F; a in F*) . The main result is that in the case of finite dimension, R = End(F, K F); note that KF} = k in F | forall x,y in F: (x + y)k = xk + yk is a subskewfield of F and (F,K F) is a right vector space. Then let F be a nearfield of dimension 2 over K F = mathbbFP . By a theorem of A. Hurwitz, End(F, K F) or the ring of all (2,2)-matrices over P is isomorphic to mathcalO} /p mathcalO . This enables us to realize every proper nearfield with p 2 elements by means of congruences in mathcalO just as in the introductory example with p = 3. © 2008 Birkhäuser Verlag Basel/Switzerland.
