(/\ i)+ p # 2 ,or ->- (/\ p is always surjective = 2 and we are in case (i) with the involution acting non-trivially on the center of Mn(:IF r) , or 2 in case (ii), then p = 2 and we are is surjective. Proof: (a) We note that if we are in case (ii), or in case (i) with A= A~ A = v + v for some surj ective, we may pick If we are in case (i), p the center of M (:IF ) n 2r v' v E: /\ with i.
1: to and Let < a, b > be a division algebra D, then the characteristic polynomial k(x) is a degree Note also that on k N(x) 2 extension of fx(Y) k. of x x f!. k belonging is irreducible k(x) - k("(x- ;tr(x)) . 1 1 T(X - ztr(x)) = - (x - ztr(x)) . regarded as a quadratic form on D , and k We now make some observations < a,b >. Clearly, if 43 x = a + Si + yj + oij then so as a quadratic form space and k < a ,b > is a division algebra if" and only i f < 1 >1 < a >1 1 has no isotropy vectors. Now assume that form Nlk(x) = 1 T(W) < v,w > we see that the is ieVT(w) + WT(V» Thus, i f (2) if = -w 1 v = x - Itr(x) , then Let a = N(x We sayan involution on k is two-dimensional.
Algebraic Topology Waterloo 1978: Proceedings of a Conference Sponsored by the Canadian Mathematical Society, NSERC (Canada), and the University of Waterloo, June 1978 by G. Carlsson, R. James Milgram (auth.), Peter Hoffman, Victor Snaith (eds.)