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K(R) two or longer. S ~ T with that 0 ~k K. If not, then the shortest nontrivial K-relation . + Rk(R) some V m the order [A:K) sum of are linearly independent over ,R} Sk(S) + Tk(T) be chosen G(K/F) Then V E V. separable, > V is the F-linear transformation Tk : V -> Tk k E K, E Set (ii) (i), V of (but no longer automorphisms necessarily), where any T and 65 - For a finite extension of fields LEMMA. 10. 10. -1 -1 -1 )a(S,S ) aO,U -1 = F; is simple; K\= eK c A CqK) = (vii) is a maximal K; [K:F]2.

With 0 ~ C € K* k €' K with k ~ kS. But then 1 ~ S G. Thus there exists a = S,T is a contradiction. contain a nonzero element of smaller length, a contradiction. a = u(S)k € I with 0 ~ k € K1'. But Thus ~(a) = 1 and k-1 u(S)-1 u(S)k T b(S,T) = c(~~~;~T)a(S,T) k. kS with then by the length reduction argument ~ 2, K* and € .. a € I 0 element +... c is of the form a and b € F. 0 If (iv) nonzero with 0 +... € center A with u(S)c suppa = {l}, S = 1, F - ak = a = u(l)a(l,l) that (3). + b(','» :: B. and u(R)K where \ (or associated, or similar)--deno ed by there exists a function c('): 1 b(',') - a("'1' > K*, if or equivalently, \ I u(S)u(T) = u(ST)a(S,T), w(S)w(T) = w(ST)b(S,T) S,T,R € G.

Is a finite separable algebraic for the basis of a right K-vector space associativity among all products of three basis elements. 4. select symbols not in The order G (KI F)! 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».

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