By De Simone A., Mundici D.

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E k[x] then b2 = b1tc where c E k and (t,n) = 1. 41. degreep. Show that if K = k(b1 ) = k(b 2 ), where bi is a root of the irreducible polynomial xP- x- ai E k[x], then b2 = nb1 + c where 0 < n <; P - 1 and CE k. (a,a)(T,b) = (aT,j(a,T)'r(a)b). Show that G' is a group with respect to this operation. Show that {(l,a) a E A} is a normal subgroup A' of G' with A~ A' and that G' fA' G. The group G' is called a group extension of A by G. 34. Lethe Z 2(G,A) and let H' be the group extension of A by G determined by h.

Case 1. lxl > 1. Letf(x) E k[x], + atx + · · · + anxn, < la1x;l for i < j and so lf(x)l = lx"f = lxln = d-deg/(x>, an d = ¢: 0.

We can now state and prove the principal result of this section. I I I Set G' = {a E G A' = {a E A Icp{a,o) = 1 for all o E G}. and THEOREM H ~ ~ KH = k(Hlfn). Furthermore, G(KHfk)"'"' H/k*n so that [KH:k] = (H:k*n). Proof. The proof will be given in several steps. Step 1. Let K be a finite Abelian extension of exponent n of kin C and let G = G(K/k). Let A = AK ={a E K*J an E k*}. cp(a,a) f = 1 for all a E A} Since G/G' is finite we have G/G'"'"' A/A', by Theorem 13. We have A' = {a E A aa(a)-1 = 1 for all a E G} = 14.

### A Cantor-Bernstein Theorem for Complete MV-Algebras by De Simone A., Mundici D.

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