“…Set Tk = 4t(zf», zf ; F') ®F, APk(zf , zf ; F') ®F,-*F. A^z^ , z%F>), and let T -T2 ®f ■ ■ ■ ®f' Tt. Then with respect to the (z¡ , ... , z2^)-adic valuation on F', we see that T is a totally ramified division algebra over F' of index m/pßl by [JW,Corollary 2.6], and T has exponent n/p"] by [TW,Theorem 4.7(i) that L is a maximal subfield of T. Since T is totally ramified over F', so is L; hence L/F' is Galois [TW,Proposition 1.4(iii)]. Thus N is normal in G. Since gcd(|7V| , \G/N\) = 1, the group G is a semidirect product N » 77 by the Schur-Zassenhaus theorem [R, p. 149]; in particular, G contains a subgroup 77 with \G:H\= pß'.…”