By Paul J. McCarthy

*Math.*studies. Over two hundred workouts. Bibliography.

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Thus (F'(a),p") E 6 and is strictly greater than (F', p'), which is a contradiction. Thus we do have F' = K and, by Proposition 1, p' is a k-automorphism of K, and so is the desired T. II THEOREM 20. The field K is a Galois extension of k, that is, k is the fixed field of G(K/k ). Proof Let a be an element of the fixed field of G(K/k). As in the proof of Proposition 1 there is a finite normal extension L of k with k s; L s; K and a E L. Let a E G(L/k). By Proposition 2 there is an element T E G(K/k) such that T(b) = a(b) for all bEL.

Show that A' is, in a natural way, a (G/H)-module. , C 2(G,A) by (A/)(a,T) = f(aH,TH). Show that A is a homomorphism which maps Z 2(G/H,A') into Z 2(G,A) and B 2(G/H,A') into B 2(G,A) and so induces a homomorphism A*: H 2(G/H,A') ... H 2(G,A). 37. Let K be a finite Galois extension of k, G = G(K/k), F a normal I 1"-J I Section 7 42. : k*~ are finite. Show that the same hold for H1H2 and H1 n H2 and thatKH 1 H, KH KH and KH 1 ,.... H, = KH 1 n KH,· 4J. Let K be a cyclic Kummer extension of the field k of Theorem ~4.

H, = KH 1 n KH,· 4J. Let K be a cyclic Kummer extension of the field k of Theorem ~4. Show that if K = KH and if ak*" is a generator of H/k*" then K = k(\Y a). Section 8 44. Fill in the details of the proof of Theorem 15. 45. Let k be a quasi-finite field of characteristic p. Show that (k+: &k+) = P· Sections 10, 11, and 12 46 Let K be a Galois extension of k and let k £ L £ K. Show that the topoiogy induced on G(K/L) by the Krull topology of G(K/k) is the Krull . topology of G(K/L). 47. Let K be a Galois extension of k.