A concrete approach to division rings by John Dauns

By John Dauns

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194, Theorem zJ). of the Galois group : u(S)K over F left elementwise fixed by every G, ca : c} : F. 44, Theorem 15; E A ,. S,T,R u(S)(u(TR)a(T,R» : u(STR)a(S,TR)a(T,R); R (u(S)u(T»u(t) : (u(ST)a(S,T»u(R) : u(STR)a(ST,R)a(s,T) . such ([Artin Now the additional hypothesis that the finite separable extension F A This algebra is called a crossed product and will be denoted by all the ingredients necessary to form it c K A : (KI F, a ( , 0 ) ) 0 or A : (KIF, a(S,T». 7. a : Each element lu(S)a(S) where a Alternatively, m(x) : (x-61)(x-6Z) ...

R} transformations v(Tk) which maps G(K/F) = {S,T,... PROOF. = (vT)k for all is necessarily of length where = 0, k(S) ~ 0 E K F Since [K:F]. 12. - For each is G(K/F) = K is finite, normal, and c and k(T) ~ O. Thus for S E G, u(S)K c A = [K:F) T (i) . A Consequently m = if bl"" I j ,;m} ,; A THE DIMENSIONS OF A CROSSED PRODUCT. F finite, normal, separable field extension m= = (iU [A:F) PROOF. u{S),u(T),... = (i) ,u(R) [K:F) c K For a ~ over F satisfy = [A:F) = m2. + u(R)K. unknowns. MAIN THEOREM OF CROSSED PRODUCTS.

1 et n be any pos1t1ve 1nteger n = 0,1 ,... an arbitrary integer k = Otl,t2,... For any S(n) = Sn S(O) = S 0 = SS ... 2 will be maintained for the special case of crossed products. 1. baT; in general are not isomorphic and following are a bTO; as different. 6). 2 has two totally n*n to be denoted by The two groups III-2. will 1 € G. as will be a division algebra. 2. 14), th~ cyclic algebra D fixed. be simply The set G It is I;s. (ko(p-l» 49], form the fields sP = t, Then Q[I;](t) = 1Q1(I;,t)c K = F[s].

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