Abstract :
Letpbe an odd prime andna positive integer and letkbe a field of characteristic zero. LetK = k(w) withwpn = a kwhereais such that [K : k] = pnand letrdenote the largest integer between 0 andnsuch thatK ∩ k(ζpr) = k(ζpr), where ζprdenotes a primitiveprth root of unity. The extensionK/kis separable, but not necessarily normal and, by Greither and Pareigis, isH-Galois withHa -Hopf algebra form of a group ringkNwhere is the normal closure ofK/k.His said to bealmost classicalifN < Gal( /k). The result is that ifr < nthen there areprHopf Galois structures onK/kfor which the associated groupNis cyclic of orderpn. Of these,pmin(r, n − r)are almost classical and the rest are non-almost classical. Whenr = n, there arepn − 1H-Galois structures for whichN Cpnof which only one is almost classical. Finally, we show that these are the only structures possible. That is, for this class of extensions,N mustbe cyclic.
