Descent Calculations for Hopf-Galois Extensions of Fields

Descent Calculations for Hopf-Galois Extensions of Fields Timothy Kohl Department of Mathematics and Statistics Boston University Boston, MA 02215 tkohl@bu.edu Abstract A Hopf-Galois structure on a field extension K/k consists of a k- Hopf algebra H together with an explicit action of H on the overfield K. The elucidation of those field extensions and the Hopf algebras which can arise was first given by Greither and Pareigis. We detail some explicit examples of these Hopf algebras and actions, including those for which the extension is already Galois in the usual sense. 1 Preliminaries Let R be a commutative ring and A a commutative R-algebra, finitely generated and projective as an R-module and H a cocommutative R Hopf algebra Definition 1.1: A is an H-module algebra if there exists µ : H End R (A), 1

an R algebra homomorphism such that for h H and x, y A: µ(h)(xy) = (h) µ(h (1) )(x)µ(h (2) )(y) (i) where (h) = (h) h (1) h (2) µ(h)(1) = ǫ(h)1 (ii) We refer to condition (i) by saying that H measures A to A. Condition (ii) is simply the statement that R A H where A H = {x A µ(h)(x) = ǫ(h)x h H} which is the fixed subalgebra of A under the action of H. Definition 1.2: A/R is an H-Galois extension if A is an H-module algebra such that (1) R = A H (2) A H = End R (A) via a h (b aµ(h)(b)) µ actually extends to an algebra map from A#H End R (A) where A#H is the smash product of A with H. As an R-module A#H = A H and we denote a h by a#h The algebra structure is given by formula: (a#h)(a #h ) = (h) aµ(h (1) )(a )#h (2) h where (h) = (h) h (1) h (2). One can think of this as a kind of semi-direct product. Equivalent to µ : H End R (A) are maps µ : H A A 2

an R-module map defining an H-module structure on A and µ : A A H an R-algebra map giving an H -comodule structure on A, making it an H object With this, the following are equivalent: Definition 1.3: A/R is H Galois (1) µ yields an isomorphism µ 1 : H A End R (A) (2) µ yields an isomorphism A A A H (3) Note: We can view (3) in the following context: Notation: Spec(A) = Hom R alg (A, ) and Spec(H ) = Hom R alg (H, ) The map µ : A A H can be dualized into a map Spec(H ) Spec(A) Spec(A) making Spec(A) into a Spec(H )-set for the group scheme Spec(H ). If H is co-commutative then H is commutative and so this makes sense. If A A = A H then we call A a Galois H -object and at the level of schemes this translates into a bijection Spec(H ) Spec(A) Spec(A) Spec(A) making Spec(A) a principal homogeneous space under the action of Spec(H ). That is, the map (g, x) (gx, x) for g Spec(H ) and x Spec(A) is bijective. For those extensions which are H-Galois there is a Galois type correspondance between R-sub-Hopf algebras of H and subalgebras of A containing R. Theorem 1.4: ([CS] - Theorem 7.6) If we define for an R-sub-Hopf algebra W of H, Fix(W) = {x A µ(w)(x) = ǫ(w)x w W } = A W 3

then the map { W H sub Hopf algebra } Fix { E R E A, E an R subalgebra } is injective and inclusion reversing. The case when F ix is also surjective is of importance and will be discussed. What we will be focusing on is the application of these ideas to separable field extensions, that is R = k and A = K some extension field of k and H a k-hopf algebra. If K is a field and S is a finite set then one may construct the K-algebra K S = Map(S, K) of set maps from S to K. The basis of this consists of {e s s S} where e s (t) = δ s,t and e s e t = δ s,t e s. Now if K/k is a separable extension of fields with normal closure K with Γ = Gal( K/k) and = Gal( K/K) as diagrammed below: K Γ K k then if we let S = Γ/ (left cosets), the following is true: There is an isomorphism: φ : K K K S given by φ(x y) = g S xg(y)e g. Moreover, if we let Γ act on K K via the left tensor factor and on K S by g (λe h ) = g(λ)e gh then φ is seen to be, in fact, a Γ-map. The isomorphism itself comes about as follows. If K = k(ξ) = K = k[x]/(f) with f the minimal polynomial for ξ, then K K = K[X]/(f) = g S K[X]/(X g(ξ)) = K S Later we shall show by an explicit calculation that this map φ is onto in a particular example. Now an action of a group N by K automorphisms 4

on K S arises via an action of N on the set S. (i.e. N B = Perm(S)) Moreover, this action is Galois (in the sense of Galois theory of rings) if and only if N is a regular subgroup of B. That is, N acts transitively and the point stabilizer of any element of S is the identity. As such, the group ring (and Hopf algebra) K[N] acts on K S to make K S / K a K[N]-galois extension under this same condition on N. If we have an action of a Hopf algebra H on the field K, that is a map H K K then there is a regular subgroup N B = Perm(S) such that K H = K[N] and the following diagram commutes: ( K H) ( K K) K[N] K S K K KS where the horizontal arrows are induced from base changing the action of H on K up to K. The reason this is the case is that first, if one has that K/k is an H-Galois extension then this is equivalent to having K/k being an H -Galois object. One then looks at the co-action of H and resulting isomorphism K K = K H. If we base change up to K we get K K = K H but as observed earlier K K = K S and so K H has K S as its underlying algebra. Since K H is a Hopf algebra then so must K S. As such there is a comultiplication : K S K S K S = KS S which must arise from a map S S S making S a group which we call N. The group structure on S = N, the associativity, existence of an identity element and inverses, arise from the co-associativity, counitary and antipode properties from the Hopf albebra structure on K S. Therefore K H = KN = ( K[N]) and so K H = K[N]. The regularity of N is as observed above. Now the regularity condition alone is not sufficient if one wants to start with an action of K[N] on K S to yield an action at the level of K. The main theorem in [GP] gives the descent theoretic and enumerative criteria for 5

determing whether a given separable extension has a Hopf Galois structure. Note, if we have L/k an extension of fields, and A a k-object (algebra, Hopf algebra etc.), another such k-object B is called an L-form of A if L A = L B as the corresponding L-objects. Theorem 1.5: ([GP] Theorem 2.1) Let K/k be a separable field extension, S and B as above, then the following are equivalent: (a) There is a k-hopf algebra H such that K/k is H-Galois (b) There is a regular subgroup N B that is normalized by Γ B. The Hopf algebra in (a) is a K-form of k[n] and can be computed by means of Galois descent. If one has a regular subgroup N of B = Perm(S) which is normalized by Γ then one can start with an action of K[N] on K S and descend it (with descent data from Γ) to an action of a k-hopf algebra H on K. Consider the following two diagrams. The first depicts the lifting of an action of an H on K to an action of K[N] on K S, ( K H) ( K K) = K[N] K S K K = KS and the second, the descent of an action of K[N] on K S to an action of a k-hopf algebra H on K. K[N] K S ( ) Γ H = ( K[N]) Γ K KS K ( ) Γ Here H is the fixed subalgebra of K[N] under the diagonal action of Γ on both the coefficients K and the elements of N. Here is where the necessity for Γ to normalize N in B comes in, for the action of Γ on N is by conjugation inside B. Note also that ( K S ) Γ = K by virtue of the above stated action of Γ. Also, observe that if N is regular then any two of the following imply the third: 6

(a) S = N (b) N acts transitively (c) Stab s (N) = 1 for all s S and, of course, dim k (H) = [K : k] = S = N. Also, the embedding Γ B is via the left action of Γ on S and moreover the condition Γ normalizes N can be rephrased as Γ Norm B (N). Now as N is regular on S then we can regard B = Perm(S) as Perm(N) instead with N embedded via the left regular representation (ie λ(n)(m) = nm). The normalizer of λ(n) in B is a classical object known as Hol(N) the holomorph of N. The set of all permuations in B = Perm(N) that fix the identity of N is canonically Aut(N) and moreover we have Hol(N) = λ(n)aut(n) (in B) and, moreover, that Hol(N) = N Aut(N) as abstract groups. In fact, it is by using the criterion Γ Hol(N) at the level of abstract groups that one can rule out certain extensions from having any Hopf Galois structures all. For example, if [K : k] = p for p an odd prime and Gal( K/k) = S p then there are no Hopf Galois structures on K/k since there is no way for S p to be contained in Hol(C p ) since the latter has order p(p 1). For some extensions, there can be many different N s that one can find which give distinct actions yet are isomorphic as abstract groups. 1.1 Notes Some additional observations should be made at this stage. First, the condition that Γ normalizes N inside B neither requires nor rules out the possibility that N Γ. If N is contained in Γ then the extension is termed almost classical for it is in this case that the map Fix giving the correspondence between {sub Hopf algebras of H} and {subf ields of K} is bijective. We shall give an example of one of these below. Secondly, although this theory is used primarily as a way of putting a Galois type of structure on an extension which is not Galois in the usual sense, this does not mean that a field extension which is Galois in the usual cannot have additional Hopf- Galois structures on it. Indeed, an ordinary Galois extension K/k with G = Gal(K/k) is already Hopf-Galois with H = k[g]. So the prescence of other Hopf-Galois structures is certainly not to be ruled out. In the case 7

of such an extension, K = K and so = 1 and S = Γ. We shall give an example of this as well in the next section. 2 Examples 2.1 Separable, non-normal extension with Hopf Galois structure Our first example will be Q(w)/Q where w = 3 2 which is certainly separable but non-normal. It is Hopf Galois for a Q-Hopf algebra H we shall exhibit below. First, let us extend our view to the normal closure of this extension, Q(w, ζ) where ζ is a primitive cube root of unity. M Q(ζ) Q(w, ζ) Q Γ Q(w) Here M = σ is cyclic of order 3 where σ(w) = ζw and = δ with δ(ζ) = ζ 2 and Γ = M = M. Consider the group ring Q(ζ, w)[m] and define α 1 = σ + σ 2 α 2 = ζ 2 σ + ζσ 2 and let H = Q[α 1, α 2 ]. The action of H on Q(w) is then induced from the action of M. Specifically, α 1 (w) = ζw + ζ 2 w = (ζ + ζ 2 )w = w 8

and α 2 (w) = ζ 2 (ζw) + ζ(ζ 2 w) = w + w = 2w Note also that α 1 (1) = 2 and α 2 (1) = 2. Now the question that arises is how did we construct H in this case? The N in this case is clearly M which is certainly normalized by Γ and moreover is contained in Γ. However some explanation as to why this works is needed. As mentioned above, we would consider the fixed ring (Q(w, ζ)[n]) Γ where Γ acts diagonally on the coefficients and the elements of N. However, we can take advantage of the following descent theoretic fact that holds in the almost classical case. In the case of an almost classical extension, K/k with K = KE where E is linearly disjoint from K over k. If M = Gal( K/E) and Γ = Gal( K/k) then B contains both M and M opp, the opposite group to M (M with multiplication reversed) inside B. Both M (and consequently M opp ) are both regular subgroups that are normalized by Γ. However, M opp has one additional property that is computationally convenient from a descent point of view, it is centralized by M. What this entails is that, since Γ = M with M a normal complement to, we can descend from K to k through E and take advantage of the fact that M centralizes M opp. What does this look like? Well, since N is our M opp here, we would have: K[M opp ] K S KS ( ) Γ H = ( K[M opp ]) Γ K However, since M centralizes M opp and Γ = M then we can first descend from K to E via M and then descend from E to k via which looks like K ( ) Γ

this K[M opp ] K S KS ( ) M E[M opp ] = ( K[M opp ]) M (E K) ( ) H = (E[M opp ]) K ( ) M (E K) K ( ) where E[M opp ] = ( K[M opp ]) M by virtue of M centralizing M opp. Note also that, since K S = K K then ( K S ) M = ( K) M K = E K. The Hopf algebra that acts is the fixed ring under the action of on the smaller group ring E[M opp ]. Now in the context of the example we were looking at, since M is abelian, then M = M opp and so the Hopf algbra H we mentioned above is H = (Q(w, ζ)[m]) Γ = (Q(ζ)[M]) which is easily computed given that acts on M via δ(σ) = σ 2. As such, the elements α 1 and α 2 are generators of the fixed ring, along with 1 of course. Note, that dim Q (H) = 3. The Hopf algebra structure on H is induced from that on Q(ζ, w)m, to wit: (σ + σ 2 ) = (σ σ + σ 2 σ 2 ) ǫ(σ + σ 2 ) = 2 λ(σ + σ 2 ) = σ 2 + σ The measuring property of H on Q(w) is evident, α 1 (w w) = m(σ σ + σ 2 σ 2 )(w w) = m(ζw ζw + ζ 2 w ζ 2 w) = ζ 2 w 2 + ζw 2 = w 2 10

which works out the same if we computed it directly, to wit, α 1 (w 2 ) = σ(w 2 ) + σ 2 (w 2 ) = ζ 2 w 2 + ζw 2 = w 2 Lastly, in [K] it is shown that this is the only Hopf Galois structure possible. The result comes from an analysis of more general prime power radical extensions of the form k(w)/k where w pn k such that [k(w) : k] = p n but this case can be argued directly as follows. Any other such N would be a cyclic group of order 3 such that Γ Hol(N ) B. However, Γ = 6 = B and, moreover, Γ = N Aut(N) = Hol(N), and so Hol(N) = Hol(N ) and so N must be N itself since N is the only subgroup of Hol(N) of order 3. 2.2 Ordinary Galois extension with additional Hopf Galois structure Our second example is of also an extension of Q which is Galois in the usual sense (hence Hopf Galois by default) and also has an additional Hopf Galois structure on it which we shall calculate. Indeed it is the normal closure of our previous example! Let w = 3 2 and ζ be a primitive cube root of unity as before and let k = Q and K = Q(w, ζ) and let Γ = σ, δ where σ and δ act on w and ζ as above. Note that here K = K. The two Hopf Galois structures we shall exhibit are related in that they arise due to choices of N which are related. First, since the extension is Galois, we have that B = Perm(S) = Perm(Γ) and recall that Γ is embedded in B via the left action of Γ on S and must normalize any N which arises. This embedding is the left regular representation of Γ and since Γ clearly normalizes itself (again the action of Γ on any N is by conjugation) then one clear choice is N l = λ(γ) the left embedding of Γ in B. This leads to the other choice for N we shall examine which relates to the observation made above in the almost classical case. Specifically, if N is a regular subgroup of B then so is N opp, note again that N opp and N centralize one another in B. In this context, N opp l = ρ(γ) the right-regular representation of Γ in B = Perm(Γ) where we define ρ(γ 1 )(γ 2 ) = γ 2 γ 1 11 1.

We shall call this N r and observe that Γ = N l certainly centralizes N r and so N r will give us a Hopf Galois structure on K/k. The point to observe is that while Γ = N l centralizes N r it does not centralize itself so the action of Γ on K[N l ] versus that on K[N r ] will be different and moreover the resulting actions on K/k will be different. We shall call the resulting Hopf algebras H l and H r and observe that the descent data that yield each are different and result in non-isomorphic algebras. Specifically, H r = (K[N r ]) Γ where Γ acts (as always) simultaneously on the group elements and the coefficients. However, since Γ acts trivially on N r then we get H r == k[n r ] or simply the canonical structure arising due to the fact that K/k is already Galois. However, H l = (K[N l ]) Γ where now the action of Γ on both the group elements and scalars is nontrivial. The action of Γ on K[N l ] is straightforward but working out a basis for the fixed ring was done using the computer algebra system MAPLE. Let us examine this a bit. The group ring Q(w, ζ)[n r ] is a 36 dimensional vector space over Q and if we regard N l = Γ as σ δ where δ(σ) = σ 2 then a typical element of the group ring looks like ( b0,0,2,1 + b 1,1,2,1 zw + b 0,1,2,1 z + b 2,1,2,1 zw 2 + b 2,0,2,1 w 2 + b 1,0,2,1 w ) (σ 2, δ) + ( b 0,0,2,0 + b 2,0,2,0 w 2 + b 1,0,2,0 w + b 1,1,2,0 zw + b 2,1,2,0 zw 2 + b 0,1,2,0 z ) (σ 2, 1) + ( b 0,0,1,1 + b 1,0,1,1 w + b 1,1,1,1 zw + b 2,1,1,1 zw 2 + b 0,1,1,1 z + b 2,0,1,1 w 2) (σ, δ) + ( b 1,0,1,0 w + b 0,1,1,0 z + b 0,0,1,0 + b 2,1,1,0 zw 2 + b 2,0,1,0 w 2 + b 1,1,1,0 zw ) (σ, 1) + ( b 0,1,0,1 z + b 0,0,0,1 + b 1,1,0,1 zw + b 2,1,0,1 zw 2 + b 2,0,0,1 w 2 + b 1,0,0,1 w ) (1, δ) + ( b 1,0,0,0 w + b 0,1,0,0 z + b 2,0,0,0 w 2 + b 2,1,0,0 zw 2 + b 0,0,0,0 + b 1,1,0,0 zw ) (1, 1) with b i,j,k,l Q. Now what was done was to examine the set of relations imposed on the b i,j,k,l by forcing the above element to be fixed by all of Γ. What was found was that the typical element of H l is an expression of the 12

form: lh = ( b 2,1,1,1 ζw 2 + b 1,0,0,1 ζw b 2,1,1,1 w 2 + b 0,0,2,1 ) (σ 2, δ) + (b 0,0,2,0 + (b 0,0,2,0 b 0,0,1,0 ) ζ)(σ 2, 1) + ( b 1,0,0,1 w + b 0,0,2,1 b 1,0,0,1 ζw + b 2,1,1,1 ζw 2) (σ, δ) + (b 0,0,1,0 + ( b 0,0,2,0 + b 0,0,1,0 ) ζ)(σ, 1) + ( b 2,1,1,1 w 2 + b 0,0,2,1 + b 1,0,0,1 w ) (1, δ) + b 0,0,0,0 (1, 1) with b 2,1,1,1, b 0,0,2,0, b 1,0,0,1, b 0,0,1,0, b 0,0,0,0, b 0,0,2,1 Q. The dimension, of course, is exactly 6 which it must be since [K : k] = 6. For reference, we shall include the typical element of H r which ends up looking like this: rh = b 0,0,0,0 (1, 1)+b 0,0,2,1 (σ 2, δ)+b 0,0,2,0 (σ 2, 1)+b 0,0,1,1 (σ, δ)+b 0,0,1,0 (σ, 1)+b 0,0,0,1 (1, δ) with b 0,0,0,0, b 0,0,2,1, b 0,0,2,0, b 0,0,1,1, b 0,0,1,0, b 0,0,0,1 Q which is just a typical element of the group ring Q[N r ] = Q[Γ]. Now the actions of these Hopf algebras on K = Q(w, ζ) are demonstrably different, observe: rh(1) = b 0,0,0,0 + b 0,0,0,1 + b 0,0,1,0 + b 0,0,1,1 + b 0,0,2,0 + b 0,0,2,1 rh(w) = ((b 0,0,1,0 + b 0,0,1,1 b 0,0,2,0 b 0,0,2,1 ) ζ + b 0,0,0,0 + b 0,0,0,1 b 0,0,2,0 b 0,0,2,1 ) w rh(w 2 ) = (( b 0,0,1,0 b 0,0,1,1 + b 0,0,2,0 + b 0,0,2,1 ) ζ + b 0,0,0,0 + b 0,0,0,1 b 0,0,1,0 b 0,0,1,1 ) w 2 rh(z) = ( b 0,0,0,1 + b 0,0,0,0 b 0,0,2,1 + b 0,0,1,0 b 0,0,1,1 + b 0,0,2,0 ) ζ b 0,0,2,1 b 0,0,0,1 b 0,0,1,1 and lh(1) = b 0,0,0,0 + 2 b 0,0,2,0 + 3 b 0,0,2,1 b 0,1,2,0 lh(w) = (2 b 0,1,2,0 + b 0,0,0,0 b 0,0,2,0 ) w + 3 b 1,1,2,1 w 2 lh(w 2 ) = 3 Awb 2,1,2,1 + ( b 0,1,2,0 + b 0,0,0,0 b 0,0,2,0 ) w 2 lh(z) = (2 b 0,0,2,0 3 b 0,0,2,1 b 0,1,2,0 + b 0,0,0,0 )z 3 b 0,0,2,1 where A = w 3. Observe also that the fixed fields, K H l and K H r are both Q. Indeed, rh(1) = ǫ(rh)1, for example, which is in accord with condition (ii) in the definition of H-module algebra. The measuring property can also be verified. 13

3 Appendix Here we shall demonstrate some of the difficulty in showing, computationally, the isomorphism K K = K S in one particular case. The calculations here were carried out using MAPLE. In the second example, we had that K = K = Q(w, ζ) and consequently S = Γ. The isomorphism then is K K = K Γ where Γ = σ δ. Now as a the tensor product is over Q, a typical element of K K is of the form c i,j,k,l (ζ j w i ζ l w k ) (i,j,k,l) Recall, the map φ : K K K Γ will work as follows: φ(x y) = g Γ xg(y)e g and applied to the generic element of K K above yields c i,j,k,l ζ j w i g(ζ l w k )e g g Γ (i,j,k,l) What are computed below are the preimages under φ of the basis elements {e g g Γ} of K Γ. φ 1 (e (1,1) ) = 1 A (w2 ζw) 1 A (ζw w2 ) 1 A (w ζw2 ) 2 A (ζw2 ζw) + 1 A (w2 w) + 1 A (w w2 ) 2 A (ζw ζw2 ) 1 A (ζw2 w) + 1 (1 1) 1 (1 ζ) 2 (ζ ζ) 1 (ζ 1) φ 1 (e (1,δ) ) = 1 A (w2 ζw) + 1 A (ζw w2 ) + 1 A (w ζw2 ) + 2 A (ζw2 ζw) + 2 A (w2 w) + 2 A (w w2 ) + 2 A (ζw ζw2 ) + 1 A (ζw2 w) + 2 (1 1) + 1 (1 ζ) + 2 (ζ ζ) + 1 (ζ 1) 14

φ 1 (e (σ,1) ) = 1 A (w2 ζw) + 2 A (ζw w2 ) + 2 A (w ζw2 ) + 1 A (ζw2 ζw) 2 A (w2 w) + 1 A (w w2 ) + 1 A (ζw ζw2 ) 1 A (ζw2 w) + 1 (1 1) 1 (1 ζ) 2 (ζ ζ) 1 (ζ 1) φ 1 (e (σ,δ) ) = 1 A (w2 ζw) + 1 A (ζw w2 ) 2 A (w ζw2 ) 1 A (ζw2 ζw) 1 A (w2 w) 1 A (w w2 ) 1 A (ζw ζw2 ) 2 A (ζw2 w) + 2 (1 1) + 1 (1 ζ) + 2 (ζ ζ) + 1 (ζ 1) φ 1 (e (σ 2,1)) = 2 A (w2 ζw) 1 A (ζw w2 ) 1 A (w ζw2 ) + 1 A (ζw2 ζw) + 1 A (w2 w) 2 A (w w2 ) + 1 A (ζw ζw2 ) + 2 A (ζw2 w) + 1 (1 1) 1 (1 ζ) 2 (ζ ζ) 1 (ζ 1) φ 1 (e (σ 2,δ)) = 2 A (w2 ζw) 2 A (ζw w2 ) + 1 A (w ζw2 ) 1 A (ζw2 ζw) 1 A (w2 w) 1 A (w w2 ) 1 A (ζw ζw2 ) + 1 A (ζw2 w) + 2 (1 1) + 1 (1 ζ) + 2 (ζ ζ) + 1 (ζ 1) We can test these calculations, for example, by applying φ to one of the 15

preimages given above. ( φ(φ 1 (e (σ 2,δ))) = 2 1 ζ 1 ) ( 2 e (1,δ) + ζ + 2 ) ( 1 e (σ 2,1) + ζ + 1 ) e (σ 2,1) ( + 1 ζ 1 ) ( 1 e (σ 2,δ) + ζ + 1 ) ( 2 e (σ,1) + ζ + 2 ) e (σ 2,δ) ( 2 + ζ + 2 ) ( 1 e (1,δ) + 2 ζ + 1 ) ( 1 e (σ,δ) + 2 ζ + 1 ) e (σ 2,δ) ( 2 + ζ + 2 ) ( e (σ,1) + 1 ζ 1 ) ( e (σ,1) + 2 ζ 2 ) e (σ,1) ( + 2 1 ζ 1 ) ( e (σ,δ) + 2 ζ 2 ) ( e (1,1) + 2 ζ 2 1 3 ζe (σ 2,δ) + 2 ( 1 3 e (σ 2,δ) + 2 ζ + 1 ) e (1,1) + ( 1 ζ 1 = e (σ 2,δ) ) e (σ 2,1) ) e (σ 2,1) 4 References [CS] S.U. Chase and M. Sweedler, Hopf Algebras and Galois Theory, Lecture Notes in Mathematics No. 7, Springer-Verlag, Berlin, 16. [GP] C. Greither and B. Pareigis, Hopf Galois Theory for Separable Field Extensions, J. Algebra 106, 187, 23 258. [K] Timothy Kohl, Classification of the Hopf Galois Structures on Prime Power Radical Extensions, J. Algebra 207 18, 525 546. [KO] M-A. Knus and M. Ojanguren, Théorie de la descente et algèbres d Azumaya, Lecture Notes in Mathematics No. 38, Springer-Verlag, Berlin, 174. [P1] B. Pareigis, Descent Theory Applied to Galois Theory, technical report, Univ. California San Diego, 186. 16

[P2] B. Pareigis, Forms of Hopf Algebras and Galois Theory,Topics in Algebra, Banach Center Publications Vol. 26 Part I, 10, 75 3. 17