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1 and =:R(t). -l)-t] = '. -(z-l) -z hex) L + (t-3)(z-1)'+ " t, + (t-3)(x-l) + t. " It follows that for any Thus F[b] = :R(z). Replacement of z-l, p(x) , ) hex) then over :R(t) is a prime, there are no intermediate subfields properly between € : FJ If p(x) or larger, it follows first that is in fact the minimal polynomial of secondly b 3 Next the polynomial computed. £ F[x], Since by the above the minimal polynomial of is necessarily of degree .. - 3z+1 - tz(z-l) = 0 = II(l-z) or by za ( Z) = 1-1 I z or za = h{z), h(h(h(x») Set za{Z) y .

2C]). d zy = and K, € = copy of aB : B -> for all ek K uK = c k € K. aB That is, € G, such that K the element eK = yes) :: c z = dy Then = yc. zm = wmaB zm = v(S)v(T) = v(ST)(ub(S,T»aB applying aB v(S)-lekv(S) -1 e(kS) = = u(S)-leku(S). ) algebra and that dyeS) = for each = PROOF. v(S)(dS). S € G. ) - 1. n = m. First note that if it follows that that F c K be regarded F m as is a finite, Set over = l}; and that v tions factor set F. ) :: 1 (K/F,l) formed in a natural way ~ay S € G and S, RCk) : V v as an m-dimensional vector space k € K :: (K/F,cr,a) = (K/F,cr,b) :: B <=--=> b = aN(c) v € V.

175, Theorem 4J). 5. DEFINITION. {S E G I a(S) K over F is a > K* = K\{O} / '--'- I runs over is uniquely of the form G while a(S) can be regarded as a function supp a suppa: S a E A a(S) f O}I , a of -I-O}. w E K. a : G > K. 0, to be . The number of elements support ! of a is the length of a. 8. 9. a. 63 u(S)B = u(l)a(l,l) The next-to-the-last = e. Special choices of elements in the equation equation shows that also defining a factor set yield the following identities valid for all R € G. and S -1 -1 -S(-l) Bu(S) = u(S )a(S ,S) -S(-l) a(l,l) u(S) = e.

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