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С . Greither

power . Hence и cannot be a parameter, so both u, v sltq normalized, and this forces ord(w-l) = ord(t)-l), and we get ord(w-l) = pf-|-l by Lemma 2.2 since (9^ is integrally closed. This proves one imphcation of the second statement. The reverse implication is a direct consequence of 2.2.

Remarks , (a) The above criterion was essentially also proved by Childs. It is straightforward to deduce also Childs' ramification criterion when (9^ {ЦК mer extension of degree p) is Hopf Galois from 2.3. The main ingredient is the following: If L = iC(w^^^), w a normalized unit with OYà{w—l) = d<pe\ then the ramification number ЬЩК) is given by pe' — d (Wyman 1969, Theorem 12). We omit all details since this is covered by 2.4. below.

( b ) For Cp^jR, we know (1.3.1) that H^ and Н^. are dual to each other. Hence, for L as in 2.3, Gj^ is an H^-Galois extension iff it is an Hj,-Galois algebra. If one writes Hi> = R[n~''{T—\)'] and uses the identifications of 1.3.1, then т operates on (9j^ as it should, namely т{и^'Р) = Ср-и^^^.

Our next aim is to consider Galois extensions L/K with group Cp in the case Cp not necessarily in R. Thus, L is a X^^-Galois extension (or, what amounts to the same, a XC^-Galois algebra) of K. The correct question (given our vention) is here: When is Ф^ an Яf-Galois extension, or, what amounts to the same, an ЯJ-Galois algebra? The result is

Theorem 2.4 Let L/K be Galois of degree p as above, S = (9i^. Then the following conditions are equivalent:

( i ) S/R is an H-Galois algebra for some Hopf order HaKCp (necessarily H

is some Я;, 0 g г ^ e') ;

( ii ) TTi^^f.(S) is the p — l'st power of an ideal of R, and the ramification number

t { L / K ) does not happen to be pe' ;

( iii ) disc(S/jR) is the p-th power of an R-ideal;

( iv ) The ramification number t = t{L/K) is congruent — 1 mod p.

Proof Let us first remark that it seems to be difficult to prove 2.4 by reduction

to the Kummer case (i.e. to 2.3 or Childs' result).

( iii ) o ( iv ) : By Serre (1962, p. 83) the different О^/к has L-valuation (t-hl)(/?-l)

( with t——l being the unramified case). Hence disc (S/R) has i^-valuation

( t +1) (p— 1), and this is divisible by p iff t= — 1 mod p.

( iv ) => ( ii ) : By loc.cit. p. 84, ordK(Tr(S)) = [(t-f l)(p-l)/p]. If t=-lmodp, this

bracket equals ------(p—1), hence (ii) holds. (The forbidden case t = pe' does

not occur since pe' is not = — 1 mod p.)

( ii ) => ( iv ) : Assume ordu(Tr(S)) divisible by p—1 and let r+l=spH-r, 0^r<p.

We must show r = 0. By the formula already used a few lines ago,

ord« ( Tr ( 5 ) ) = 5.(p-l)+[r(p-l)/p],

and the last bracket has value between 1 and p —2 for r between 2 and p —1, hence it only remains to rule out the possibiHty r = l. But we have the extra hypothesis t^pe\ and this implies t not divisible by p by Wyman (1969, lary 14 and Corollary 16), hence r=f= 1.

( i ) => ( iii ) : Suppose S is an Я-Galois algebra. Then it is an Я^-Galois extension, and by 1.3 (a) and 1.5 the discriminant disc(iS'/i^) is a p-th power.