New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math.17(2011) 51–74.

Hopf Galois structures on Kummer extensions of prime power degree

Lindsay N. Childs

Abstract. Let K be a field of characteristic not p (an odd prime), containing a primitive pⁿ-th root of unity ζ, and let L = K[z] with x^pn−a the minimal polynomial ofz overK: thus L|K is a Kummer extension, with cyclic Galois groupG=hσiacting onLviaσ(z) =ζz.

T. Kohl, 1998, showed thatL|K has pⁿ⁻¹ Hopf Galois structures. In this paper we describe these Hopf Galois structures.

Contents

1. Introduction 51

2. Greither–Pareigis theory and Byott’s translation 53

3. The action ofLN^G onL 54

4. Regular embeddings forG cyclic 55

5. Regular subgroups of Perm(G) forG cyclic 57

6. Determining the K-Hopf algebra 59

References 73

1. Introduction

Chase and Sweedler [CS69] introduced the concept of a Hopf Galois ex- tension, generalizing the notion of a classical Galois extension of fieldsL|K with Galois group G. Given a field extension L ⊃ K and a cocommuta- tive K-Hopf algebraH that acts on Lmaking Linto an H-module algebra (i.e., h(ab) =P

(h)h₍₁₎(a)h₍₂₎(b), where, following Sweedler’s notation, the comultiplication ∆ :H→H⊗_KH is given on an elementh ofH by

∆(h) =X

(h)

h₍₁₎⊗h₍₂₎,

thenL is anH-Hopf Galois extension of K if the obvious map L⊗_KH→End_K(L)

induced from the module action ofH onL is 1-1 and onto.

Received January 31, 2010.

2000Mathematics Subject Classification. 12F10.

Key words and phrases. Hopf Galois extension, Kummer extension,p-adic logarithm.

ISSN 1076-9803/2011

51

LINDSAY N. CHILDS

IfLis a Galois extension ofK with Galois groupG, thenLis aKG-Hopf Galois extension ofK.

For some time after the concept was introduced, there was little interest in applying it to classical Galois extensions of fields, until in [GP87], Greither and Pareigis showed that a classical Galois extension L|K of fields with Galois group G could have Hopf Galois structures other than the classical one by KG, and showed how to transform the problem of determining the number of Hopf Galois structures onL|K into one depending purely on the Galois group G.

Subsequently, Byott [By96], extending [Ch89], found a translation of the group-theoretic problem that made the problem of counting Hopf Galois structures on L|K far more tractable. Thus most of the results on Hopf Galois structures on field extensions with given Galois groupGhave utilized Byott’s translation, in particular, [By96], [Ko98], [CaC99], [By02], [Ch03], [By04a], [By04b], [Ch05], [By07], [Ch07], [CCo07], [FCC11]. Other than the original paper of Greither and Pareigis [GP87] very few papers explicitly use the direct approach of [GP87] to determine Hopf Galois structures and even fewer explicitly describe the K-Hopf algebra and the action of the Hopf algebra on the field extension L. The most notable exceptions are papers that utilize the Kummer theory of formal groups to yield Hopf Galois structures (see [Ch00], Chapter 12), results where the Galois group has order p², p an odd prime (see [Ch96] and [By02]), and work of Kohl [Ko07], for groups Gof order 4p,p an odd prime.

For Galois extensions L|K of local number fields with Galois group G, a classical problem in local Galois module theory is to understand the val- uation ring S of L as a module over the group ring RG, where R is the valuation ring ofK. E. Noether showed that S is a free RG-module if and only if L|K is tamely ramified. In the wild (= non-tame) case Leopoldt showed that sometimesS is a freeAmodule whereAis the associated order of S in KG. More generally, if L|K is an H-Hopf Galois extension and A, the associated order of S in H, is an R-Hopf order in H, then S is A-free [CM94]. Byott ([By97a], [By97b], [By99], [By00], [By02]) has constructed examples of wild Galois extensions L|K of local fields with Galois group G where S is not a free module over the associated order in KG, but the associated order in some other H-Hopf Galois structure on L|K is an R- Hopf order inH, and hence S is free over that associated order . Thus the existence of Hopf Galois structures other than the classical one for Galois extensions of local fields adds an array of new possibilities for the study of local Galois module structure.

The purpose of this paper is to study the Hopf Galois structures on a cyclic extension L|K of order pⁿ, where p is an odd prime. It has been known since [Ko98] that there are pⁿ⁻¹ Hopf Galois structures onL|K, but except for n≤2 the K-Hopf Galois structures have not been described. In this paper, when L is a cyclic Kummer extension of K of degree pⁿ, p an

odd prime, we describe the K-Hopf algebras H as K-algebras for each of thepⁿ⁻¹ Hopf Galois structures, and describe how the elements ofH act on L.

Acknowledgements. My thanks to the Mathematics Department at Vir- ginia Commonwealth University for its hospitality while this research was conducted, and to a referee for numerous helpful comments on a previous version of this paper.

2. Greither–Pareigis theory and Byott’s translation

In order to pass from results obtained via Byott’s translation to a more explicit description of the Hopf Galois structures, we need to examine the Greither–Pareigis and Byott results in some detail.

LetL|K be a Galois extension with Galois group G. Then the map γ :L⊗_KL→Hom_L(LG, L) :=GL

by γ(a⊗b)(σ) =aσ(b) is an isomorphism. If{x_σ :σ∈G}is the dual basis of{σ ∈G}, thenL∼=K⊗_KLmaps underγ intoGLand the imageγ(1⊗b) satisfies

γ(1⊗b)(σ) =σ(b) for all σ inG. Thus

γ(1⊗b) =X

σ∈G

σ(b)x_σ, and for τ inG,

γ(1⊗τ(b)) = X

σ∈G

σ(τ(b))x_σ

= X

σ∈G

σ(b)x_στ⁻¹

= X

σ∈G

σ(b)x_ρ(τ)(σ)

whereρ:G→Perm(G) is the right regular representation. Thus the action of G on L ∼= K ⊗L corresponds under base change to an action of G on GL = P

τLxτ making GL a Galois extension of L with Galois group G acting as permutations of the dual basis {x_σ}via ρ.

A subgroup N of Perm(G) is regular if N has the same order as G and the orbit inG of each element ofG under action byN is all of G.

Greither and Pareigis showed that given any Hopf Galois structure onL|K by a K-Hopf algebra H, thenL⊗_KH =LN for some regular subgroupN of Perm(G) acting on GL by permuting the (subscripts of the) dual basis {x_σ}. Also, N is normalized by λ(G), the image in Perm(G) of G under the left regular representation λ of G, λ(σ)(τ) = στ. Conversely, each regular subgroup N of Perm(G) defines an LN-Hopf Galois structure on GLby permuting the dual basis{x_σ}, and ifN is normalized byλ(G), then

LINDSAY N. CHILDS

that Hopf Galois structure descends to an (LN)^G-Hopf Galois structure on (GL)^G∼=L, where Gacts onGL by

τ(ax_σ) =τ(a)x_λ(τ)(σ) forτ, σ inG,ainL, and Gacts on LN by

τ(aη) =τ(a)λ(τ)ηλ(τ)⁻¹

forτ inG,ηinN andainL. Thus there is a bijection between Hopf Galois structures onL|K and regular subgroups of Perm(G) normalized by λ(G).

Byott’s translation works as follows. Given a Galois extension L|K with Galois group G, let N be a group with the same cardinality as G. Inside Perm(N) is Hol(N), the normalizer ofλ(N). Then Hol(N) =ρ(N)·Aut(N).

Each homomorphism β : G → Hol(N) so that β(G) is a regular subgroup of Perm(N) yields a Hopf Galois structure on L|K, as follows: Given β, define the function b : G → N by b(σ) = β(σ)(e), where e is the identity element ofN. Sinceβ(G) is a regular subgroup of Perm(N),bis a bijection.

Thenbdefines an isomorphism between Perm(N) and Perm(G) by the map that sends π in Perm(N) to C(b⁻¹)(π) =b⁻¹πb in Perm(G). So define an embedding α:N →Perm(G) by

α(η)(σ) =b⁻¹(λ(η)(b(σ)))

forσ inG,η inN. As Byott [By96] showed, two regular embeddingsβ, β⁰: G → Perm(N) define the same regular subgroup α(N) of Perm(G), hence the same Galois structure onL|K, iff there exists an automorphismγ ofN so that β⁰(σ) =C(γ)(β(σ)) =γβ(σ)γ⁻¹ for allσ inG.

This translation of the problem of finding regular subgroups of Perm(G) normalized byλ(G) has made the problem of determining Hopf Galois struc- tures onL|K somewhat easier by first, splitting the problem into a number of separate problems, one for each isomorphism class of groups N of the same cardinality asG, and second, replacing the problem of finding regular subgroups of Perm(G) normalized by λ(G) by finding regular embeddings of Ginto Hol(N), typically a much smaller group. As a result, most of the results on Hopf Galois structures on field extensions with given Galois group Ghave utilized Byott’s translation, as noted above.

Moving from results using the Byott translation to explicitly describing the Hopf Galois structure involves two obstacles: first, using the function b (rarely a homomorphism) to translate from a regular embedding β : G → Hol(N) to the corresponding embedding α :N → Perm(G), and secondly, descending the action of α(N) onGL to an action ofL(α(N))^λ(G) on L.

3. The action of LN^G on L

Let L|K be a Galois extension of fields with Galois groupG of order n, let N be a group of order nand suppose β :G→ Hol(N)⊂Perm(N) is a regular embedding. Then N gives rise to a Hopf Galois structure on L|K by the K-Hopf algebra H=LN^G∼=Lα(N)^λ(G). We describe this action.

Proposition 1. Let L|K, G, N, β, H be as above. Then H acts on L as follows. For ξ=P

η∈Nsηη in H,sη in L anda in L, ξ(a) =X

η

sηb⁻¹(η⁻¹)(a).

Proof. The regular embedding β :G→ Hol(N) yields the regular embed- dingα:N →Perm(G) via

α(η)(σ) =b⁻¹(λ(η)(b(σ)).

Forain L, the image ofainGLis P

σσ(a)x_σ, and

ξ X

σ∈G

σ(a)x_σ

!

=X

η

s_ηη X

σ

σ(a)x_σ

!

=X

η,σ

sησ(a)x_α(η)(σ). Since H maps GL^G to itself, ξ(P

σσ(a)x_σ) has the form P

σσ(c)x_σ, the image inGLof an element cof L. Thus forainL,ξ(a) =c, the coefficient of x₁ in the last expression (where 1 is the identity element ofG).

Nowb:G→N is a bijection that maps 1 inGto the identity elemente of N, sinceβ is a homomorphism. So ,

1 =α(η)(σ) =b⁻¹(λ(η)b(σ)) iff

ηb(σ) =e, iff

σ=b⁻¹(η⁻¹).

So the coefficient of x1 is

ξ(a) = X

(η,σ),α(η)(σ)=1

s_ησ(a)

=X

η∈N

s_ηb⁻¹(η⁻¹)(a).

Once we find a set of generators of H =LN^G, we may use the mapb⁻¹ to describe the action ofH onL as in Proposition 1.

4. Regular embeddings for G cyclic

LetLbe a Galois extension ofK with Galois groupGcyclic of orderpⁿ, p an odd prime. Kohl [Ko98] showed that if L|K is H-Hopf Galois, then the K-Hopf algebraH has associated groupG: that is, L⊗_KH ∼=LG. In other terms, if there is a regular embedding ofGinto Hol(N) forN a group of orderpⁿ, thenN ∼=G. Thus if we wish to find Hopf Galois structures on L|K, it suffices to seek regular embeddings β of Ginto Hol(G).

LINDSAY N. CHILDS

Now Hol(G)∼=GoAut(G)∼=Z/pⁿZo(Z/pⁿZ)^×, where we view elements of Hol(G) as of the form (a, g) with a, g integers modulo pⁿ and g coprime to p. Since G embeds in Perm(G) by the right regular representation ρ, (a, g) acts on h inG by (a, g)(h) = (gh−a), and so the multiplication on Gis (a, g)(a⁰, g⁰) = (a+ga⁰, gg⁰). It is routine to verify that (a, g) has order pⁿ iff a is coprime to p and g ≡ 1 (mod p). Thus for G =hσi the regular embeddings β : G → Hol(Z/pⁿZ) have the form β(σ) = (a,1 +dp) for a coprime to p.

Proposition 2. Up to equivalence, the pⁿ⁻¹ equivalence classes of regular embeddings β :G→Hol(G) are represented byβ satisfying

β(σ) = (−1,1 +dp) for d modulopⁿ⁻¹.

Proof. Given β, β⁰ : G→ Hol(G), β ∼ β⁰ ifβ⁰ =C(γ)β for γ in Aut(G).

Now (Z/pⁿZ)^× ∼= Aut(G) via g 7→ γ_g, left multiplication by g. For h in G=Z/pⁿZandβ(σ) = (a,1+dp) in Hol(G) =ρ(G)·Aut(G)∼=GoAut(G),

γgβ(σ)γ_g⁻¹(h) =γg(a,1 +dp)γ_g⁻¹(h)

=γ_g(a,1 +dp)(g⁻¹h)

=γg((1 +dp)g⁻¹h−a)

=g((1 +dp)g⁻¹h−a)

= (1 +dp)h−ga

= (ga,1 +dp)(h).

For eachacoprime top, there is somegso thatga≡ −1 (modpⁿ). Sincedis unaffected byγ_g, each choice of dmodulopⁿ⁻¹ yields a different equivalence

class.

For later use we note:

Lemma 3. Forg in (Z/pⁿZ)^×, (−1, g)λ(a)(−1, g)⁻¹ =λ(ga).

Proof.

(−1, g)(g⁻¹, g⁻¹) = (−1 +g(g⁻¹), gg⁻¹) = (0,1) and so for all h inG,

(−1, g)λ(a)(−1, g)⁻¹(h) = (−1, g)λ(a)(g⁻¹, g⁻¹)(h)

= (−1, g)λ(a)(g⁻¹h−g⁻¹)

= (−1, g)(a+g⁻¹h−g⁻¹)

=g(a+g⁻¹h−g⁻¹) + 1

=ga+h

=λ(ga)(h).

5. Regular subgroups of Perm(G) for G cyclic

As noted earlier, given a regular embedding β :G→ Hol(N), we obtain the corresponding regular subgroupα(N) of Perm(G) by using the bijective function b:G→ N defined by b(σ) = β(σ)(0), where 0 is the identity ele- ment ofN. Thenbdefines an isomorphism between Perm(N) and Perm(G) by the map that sendsπin Perm(N) toC(b⁻¹)(π) =b⁻¹πbin Perm(G). For Ga cyclic group (written additively), this isomorphism yields an embedding α:N →Perm(G) by

α(θ)(σ) =b⁻¹(λ(θ)(b(σ))) =b⁻¹(θ+b(σ))

for σ in G, θ in N. The subgroup of Perm(G) corresponding to β is then α(N), and the action ofLN^G onL is described byb⁻¹, as in Proposition 1.

ForG cyclic of orderpⁿ,N ∼=Gand the groupM =α(G)⊂Perm(G) is generated byη =α(1), the permutation that sendsq inGtob⁻¹(b(q) + 1).

In particular, for q=b⁻¹(k), we have

η(b⁻¹(k)) =b⁻¹(bb⁻¹(k) + 1) =b⁻¹(k+ 1).

Thus the cycle description of the generator η ofα(G) in Perm(G) is (b⁻¹(1), b⁻¹(2), b⁻¹(3), . . .).

To find b⁻¹:N →G, we have

Proposition 4. Let G = Z/pⁿZ, p odd, and let β : G → Hol(G) with β(1) = (−1, g) for g= 1 +dpas in Proposition 2. Then for t,s in G,

β(t) =

−g^t−1 g−1, g^t

, b(t) = (1 +dp)^t−1

dp b⁻¹(s) = log_p(1 +sdp)

log_p(1 +sp) where log_p(y) is thep-adic logarithm function.

Proof. If β:G→Hol(G) withβ(1) = (−1, g) for g= 1 +dp, then β(t) = (−1, g)^t= (−(1 +g+g²+. . .+g^t−1), g^t) =

−

g^t−1 g−1

, g^t

. so

b(t) =β(t)(0) = (−1, g)^t(0).

LINDSAY N. CHILDS

Forg= 1 +dp,

(−1, g)^t= −Pt r=1

t r

(dp)^r

dp ,

t

X

r=0

t r

(dp)^r

!

= −

t

X

r=1

t r

(dp)^r−1,

t

X

r=0

t r

(dp)^r

! ,

= −(1 +dp)^t−1

dp ,

t

X

r=0

t r

(dp)^r

! . So

s=b(t) =β(t)(0) = (1 +dp)^t−1

dp .

Thust=b⁻¹(s) where

1 +sdp= (1 +dp)^t.

Solving for t is the same as solving the discrete logarithm problem in the cyclic group (1 +pZ)/(1 +pⁿZ).

Forx a multiple ofp, thep-adic logarithm function is log_p(1 +x) =

∞

X

i=1

(−1)ⁱ⁻¹xⁱ i .

Forx, yboth multiples of p, log_p((1 +x)(1 +y)) = log_p(1 +x) + log_p(1 +y), and log_p : (1 +pZ)/(1 +pⁿZ) → pZ/pⁿZ is bijective [Co00, 4.2.7, 4.2.8].

Thus from 1 +sdp= (1 +dp)^t we obtain

log_p(1 +sdp) = log_p((1 +dp)^t) =tlog_p(1 +dp) and so

t=b⁻¹(s) = log_p(1 +sdp)

log_p(1 +dp) .

Example 5. Letp= 3,G=Z/9Z and d= 1, sog= 4. Then fort≥0, β(t) = (−1,4)^t,

so

b(t) = (1 +p)^t−1

p =t+t(t−1) 2 p+. . .

≡t+ 6t(t−1) (mod 9).

Thus

b(3r) = 3r b(1 + 3r) = 1 + 3r b(2 + 3r) = (2 + 3r) + 3,

hence

b⁻¹(3r) = 3r b⁻¹(1 + 3r) = 1 + 3r b⁻¹(2 + 3r) = (2 + 3r)−3.

As an element of Perm(G), the generator η ofα(G) has cycle description (0,1,8,3,4,2,6,7,5).

Since λ(1) = (0,1,2,3,4,5,6,7,8), one may verify easily that λ(1)ηλ(1)⁻¹ =η⁴.

6. Determining the K-Hopf algebra

For the remainder of the paper we assume that K contains a primitive pⁿ-th root of unity ζ, p odd, and L is a Kummer extension of K of order pⁿ, so thatL=K[z] and the minimal polynomial ofz overK isx^pn−afor someainK.

The regular subgroupα(G) =M of Perm(G) yields aK-Hopf algebra ac- tion onL by theK-Hopf algebraH=LM^G, whose structure is determined by howG acts on L and on howλ(G) acts on M. The action of Gon L is given. In our case L= K[z] is a Kummer extension of K, so the action is transparent. The action of λ(G) on M is also straightforward.

Proposition 6. Let G = Z/pⁿZ, p odd. Suppose β : G → Hol(G) is a regular embedding withβ(1) = (−1, g)where g= 1 +dpas in Proposition 2.

Then the group α(G) = M = hηi where η = α(1) = C(b⁻¹)(λ(1)). The group M is normalized byλ(G) in Perm(G). In fact, the action of λ(G) on M is induced byλ(1)ηλ(1)⁻¹ =η^g.

Proof. We know that η=C(b⁻¹)(λ(1)), and we showed in Lemma 3 that (−1, g)λ(1)(−1, g)⁻¹ =λ(g·1).

When we translate to Perm(G) via C(b⁻¹), we obtain

C(b⁻¹)(−1, g)C(b⁻¹)(λ(1))C(b⁻¹)(−1, g)⁻¹ =C(b⁻¹)(λ(1))^g, that is,

C(b⁻¹)(−1, g)ηC(b⁻¹)(−1, g)⁻¹ =η^g,

using multiplicative notation for composition of permutations in Perm(G).

Now fort inG=Z/pⁿZ, we have

b(t) = g^t−1 g−1,

LINDSAY N. CHILDS

and

C(b⁻¹)(−1, g)(t) =b⁻¹(−1, g)b(t)

=b⁻¹(−1, g)

g^t−1 g−1

=b⁻¹

g

g^t−1 g−1

+ 1

=b⁻¹

g^t+1−1 g−1

=b⁻¹(b(t+ 1))

=t+ 1.

So C(b⁻¹)(−1, g) =λ(1). Thus

λ(1)ηλ(1)⁻¹ =η^g.

Translating to multiplicative notation for the Galois group G = hσi ∼= Z/pⁿZof the Kummer extensionL|K, letα(G) =M =hηias in Proposition 6. The action of G on LM is induced by the Galois action of G on L and the action on M by σ(η) =λ(σ)ηλ(σ)⁻¹ =η^g. Thus Proposition 6 enables us to determine the K-Hopf algebraH =LM^G acting on L.

As a K-module we can determine a set of generators of H by simply taking the sums of elements in the distinct orbits under the action of G of z^kη^l for all k, l with 0 ≤ k, l ≤ pⁿ−1. But we are interested in H as a K-algebra, so our objective in the remainder of the paper is to obtain an economical set of generators of H as a K-algebra.

As an initial model, we first do the known case whereG=hσiis cyclic of order p².

6.1. G of order p². We suppose L is a cyclic Kummer extension ofK of order p² with Galois group G= hσi. Thus K contains a primitive p²-root of unity ζ, andL=K[z] with σ(z) =ζz, where the minimal polynomial of zoverK isx^p2−a. LetM =hηibe cyclic of orderp² whereσ(η) =η^g with g= 1 +dp. Thenσ(η^p) =η^p, so η^p is inLM^G=H. Therefore the minimal idempotents

e¹_s = 1 p

p−1

X

i=0

ζ^−spiη^pi

of K[hη^pi] for s = 0, . . . , p−1 are fixed by G. These idempotents satisfy η^pe¹_s =ζ^spe¹_s. It follows that

σ(z^−sdpe¹_sη) =ζ^−sdpz^−sdpe¹_sη^1+dp

=ζ^−sdpz^−sdpe¹_sη^dpη

=z^−sdpe¹_sη.

So following the construction of Greither [Gr92], we let a_v =

p−1

X

s=0

v^se¹_s wherev=z^−dp. Then

σ(avη) =avη.

Proposition 7. H=LM^G =K[η^p, a_vη] where v=z^−dp . Proof. Since H has dimensionp² overK and K[η^p] =Pp−1

s=0Ke¹_s is a sub- algebra of H, it suffices to show that fors= 0, . . . , p−1, K[η^p, avη]e¹_s has dimensionp overKe¹_s. Now

K[η^p, avη]e¹_s=K[avη]e¹_s =K[z^−sdpη]e¹_s,

the elements {(z^−sdpe¹_sη)^r : 0 ≤r < p} are linearly independent over Ke¹_s, and

(z^−sdpe¹_sη)^p =z^−p2sdζ^spe¹_s

is inKe¹_s. Thus for each s,K[η^p, a_vη]e¹_s has dimensionp over Ke¹_p. Hence K[η^p, avη] has dimensionp² overK and is a subalgebra ofH, hence is equal

toH.

This result is in [Ch96, Section 1].

6.2. G of order p³. To preview the main result, Theorem 12 below, we write down the result for L|K a Kummer extension with Galois group G, cyclic of orderp³.

InKG we have the idempotents e¹_t = 1

p²

p²−1

X

j=0

ζ^−pjtη^pj for 0≤t < p², and

e²_t = 1 p

p−1

X

j=0

ζ^−p2jtη^p2j

for 0 ≤ t < p. Then e²_t is G-invariant for all t, and e¹_t is G-invariant if p dividest.

Forσ(η) =η^1+dpwith (p, d) = 1, then, analogous to the Greither element for thep²-case, we let

g_1,sp=z^−sdp2e¹_spη, for 0≤s≤p−1, and g1,s=

p−1

X

i=0

σⁱ(z^−sdpe¹_sη) for 1≤s≤p−1,

LINDSAY N. CHILDS

the sum of the conjugates of z^−sdpe¹_sη, and let h=

p−1

X

s=1

g1,s+

p−1

X

s=0

g1,sp. Then Theorem 12 shows that

LM^G=K[h, η^p2, e²₀η^p, e¹₀η].

Forσ(η) =η^1+dp2, Theorem 12 specializes to show that LM^G =K[h, η^p, e²₀η]

where

h=

p²−1

X

s=0

g1,s=

p²−1

X

s=0

z^−sdp2e¹_sη.

6.3. Gof order pⁿ. LetL=K[z], a cyclic Kummer extension of fields of order pⁿ with Galois group G = hσi as at the beginning of Section 6. Let M =hηibe the regular subgroup of Perm(G) normalized byGcorresponding to the regular embedding β : G → Hol(G) where β(G) = h(−1,1 +dp^ν)i with (d, p) = 1 and ν ≥1. Then the corresponding K-Hopf algebra acting on L is LM^G, where G acts on L via the Galois action and acts on M by σ(η) =η^1+dpν .

Idempotents. To obtain a set of algebra generators forLM^G, we first look at the idempotents of KM.

For r= 0, . . . , n−1 and t modulo p^n−r we have the pairwise orthogonal idempotents ofK[η^pr],

e^r_t = 1 p^n−r

p^n−r−1

X

i=0

ζ^−tpriη^pri. Then

η^pre^r_t =ζ^prte^r_t so for all k≥r,

η^spke^r_t =ζ^stpke^r_t, and

1 =

p^n−r−1

X

t=0

e^r_t.

More generally, the idempotents of K[η^pr+1] decompose in K[η^pr] as:

Lemma 8. Forr = 0, . . . , n−2,

p−1

X

k=0

e^r_t+kpn−r−1 =e^r+1_t .

Proof.

p−1

X

k=0

e^r_t+kpn−r−1 = 1 p^n−r

p−1

X

k=0 p^n−r−1

X

i=0

ζ^{−pri(t+kpn−r−1)}η^pri

= 1

p^n−r

p^n−r−1

X

i=0

η^priζ^−prit

p−1

X

k=0

ζ^−ikpn−1. and the last sum = 0 if i6≡0 modulop, and =pifi=pj. So

p−1

X

k=0

e^r_t+kpn−r−1 = 1 p^n−r−1

p^n−r−1−1

X

j=0

ζ^−pr+1jtη^pr+1j =e^r+1_t . Corollary 9. For all k >0 and all s, t, r,

e^r_se^r+k_t = 0 if t6≡s (modp^n−r−k)

=e^r_s if t≡s (modp^n−r−k).

Proof. Since e^r+1_s =e^r+1_s e^r+1_s , Lemma 8 gives

p−1

X

k=0

e^r_s+kpn−r−1 =

p−1

X

k=0

e^r+1_s e^r_s+kpn−r−1. Multiplying both sides by e^r_s, we get

e^r_s=e^r+1_s e^r_s

by pairwise orthogonality of the idempotents {e^r_s : s = 0, . . . , p^n−r−1}.

Since the subscript t of e^r+1_t is defined modulo p^n−r−1, the result is then clear fork= 1. For generalk >0, we can use the casek= 1 to write

e^r_s=e^r_se^r+1_s · · ·e^r+k_s . Then e^r+k_t e^r_s=e^r_s iff

e^r+k_s =e^r+k_t , iff

s≡t (modp^n−r−k),

and equals 0 otherwise.

The groupG=hσi acts on the idempotents by σ(e^r_t) =e^r_t(1+dpν)⁻¹,

LINDSAY N. CHILDS

where the subscript t(1 +dp^ν)⁻¹ is modulo p^n−r. To verify this G-action, the change of variables j=i(1 +dp^ν) gives

σ(e^r_t) = 1 p^n−r

p^n−r−1

X

i=0

ζ^−tpriη^pri(1+dpν)

= 1

p^n−r

p^n−r−1

X

j=0

ζ^{−tprj(1+dpν)−1}η^prj

=e^r_t(1+dpν)⁻¹.

Fort=sp with scoprime to pand for k≥0, we have σ^pk(e^r_sp) =e^r

sp(1+dp^ν)^−pk.

The subscript ton e^r_t is modulop^n−r, and fork=n−r−−ν, sp(1 +dp^ν)^−pk ≡sp (modp^n−r).

Hence

σ^{p(n−r−−ν)}(e^r_sp) =e^r_sp.

In particular, for r=n−−ν and = 0, . . . , n−1−ν, we have σ(e^n−ν−_sp ) =e^n−ν−_sp .

We may now write 1 as a sum of pairwise orthogonal G-invariant idem- potents:

Proposition 10. In LM^G, we have 1 =

n−ν−2

X

=0

p^ν−1

X

s=1,(s,p)=1

e^n−ν−_sp +

p^ν−1

X

s=1

e¹_spn−ν−1.

Proof. We just observed that all of the idempotents in the sum are fixed by G.

We have

1 =

p^ν−1

X

s=0

e^n−ν_s =

p^ν−1

X

s=1,(s,p)=1

e^n−ν_s +

p^ν−1−1

X

r=0

e^n−ν_rp .

Now by Lemma 8,

e^n−ν_rp =

p−1

X

k=0

e^n−ν−1_rp+kpν,

also a sum of pairwise orthogonal idempotents, so

p^ν−1−1

X

r=0

e^n−ν_rp =

p^ν−1−1

X

r=0 p−1

X

k=0

e^n−ν−1_rp+kpν

=

p^ν−1

X

t=0

e^n−ν−1_tp

=

p^ν−1

X

s=1,(s,p)=1

e^n−ν−1_sp +

p^ν−1−1

X

r=0

e^n−ν−1_rp2 .

Repeating this decompositionn−ν−2 more times yields the desired formula.

Generators. We want to find generators of LM^Ge^n−ν−_sp over Ke^n−ν−_sp . By analogy with the p² case, which involves the sum of Greither generators z^−sdpe¹_sη, we look at elements of the form

z^ke^r_spη^pr−1

with (s, p) = 1. For n = 2, the summands z^−sdpe¹_sη are fixed by G. For n > 2 what are fixed are sums of conjugates under the action of G. We therefore need to know what power ofσ fixes these elements.

Proposition 11. We have

σ^{pn−r−−ν}(z^{−sdp+r+ν−1}e^r_spη^pr−1) =z^{−sdp+r+ν−1}e^r_spη^pr−1. Proof. We have

σ^{pn−r−−ν}(z^{−sdp+r+ν−1}) =ζ^−sdpn−1z^{−sdp+r+ν−1} while

σ^{pn−r−−ν}(e^r_sp) =e^r_sp and σ(η) =η^1+dp. So

σ^{pn−r−−ν}(e^r_spη^pr−1) =e^r_spη^{pr−1(1+dpν)k} wherek=p^{n−r−−ν}. The exponent ofη is

p^r−1(1 +dp^ν)^{pn−r−−ν} =p^r−1+p^{n−r−−ν}p^r−1dp^ν +d⁰p^{n−r−−ν}p^r−1p^2ν

=p^r−1+dpⁿ⁻⁻¹+d⁰⁰pⁿ⁻

for somed⁰⁰. So since r≤n−−ν ≤n−−1,

σ^{pn−r−−ν}(e^r_spη^pr−1) =e^r_spη^{pr−1+dpn−−1+d00pn−}

=ζ^sdpn−1e^r_spη^pr−1.

The respective powers of ζ cancel to give the result.

LINDSAY N. CHILDS

Now we can define the generators of the K-Hopf algebra LM^G.

For 0≤≤n−ν−1, 1≤r≤n−−ν,and 1≤s≤p−1, letgr,sp be the sum of the conjugates of the Greither elements z^{−sdp+r+ν−1}e^r_spη^pr−1:

gr,sp=

p^{n−−r−ν}−1

X

i=0

σⁱ(z^{−sdp+r+ν−1}e^r_spη^pr−1).

Then gr,sp is fixed by G for all s by Proposition 11, so lies in LM^G. In particular,

g1,sp =

p^{n−−ν−1}−1

X

i=0

σⁱ(z^−sdp+νe¹_spη), while

gn−−ν,sp =z^−sdpn−1e^n−−ν_sp η^{pn−−ν−1}.

We set h to be the sum of all the sums of conjugates withr= 1:

h=

n−ν−2

X

=0

p^ν−1

X

s=1,(s,p)=1

g1,sp+

p^ν−1

X

s=0

g_1,sp^n−ν−1.

Recall that σ(η) = η^1+dpν with (d, p) = 1 and ν ≥ 1. Thus η^pn−ν is in LM^G since σ(η^pn−ν) = η^{pn−ν(1+dpν)} = η^pn−ν. Also, for r = ν, . . . , n−1, e^r₀η^pr−ν is in LM^G, forσ(e^r₀) =e^r₀ and so

σ(e^r₀η^pr−ν) =e^r₀η^{pr−ν(1+dpν)} =e^r₀η^pr−νη^prd=e^r₀η^pr−ν sincee^r₀η^pr =e^r₀.

Let

H=K[h, η^pn−ν, eⁿ⁻¹₀ η^pn−1−ν, . . . , e^r₀η^pr−ν, . . . , e^ν₀η].

Evidently, H⊂LM^G.

The main result. We show that the algebra generators of H generate all of LM^G:

Theorem 12. H =LM^G.

Proof. The idea of the proof is to take the idempotents in Proposition 10, break up K[h] into a direct product corresponding to those idempotents, and count the dimensions over K of the direct factors.

We first show that the idempotents e^n−ν−_sp appearing in Proposition 10 are inH. For = 1, . . . , n−ν−1, we have

e^n−ν−_sp =eⁿ⁻₀ e^n−ν−_sp

by Corollary 9. Since

e^n−ν−_sp = 1 p^ν+

p^ν+−1

X

j=0

ζ^−jspn−νη^jpn−ν−,

we may multiply both sides by eⁿ⁻₀ to get e^n−ν−_sp = 1

p^ν+

p^ν+−1

X

j=0

ζ^−jspn−νeⁿ⁻₀ (η^pn−ν−)^j,

a K-linear combination of powers of eⁿ⁻₀ η^pn−ν−, hence in H. For = 0, the idempotents e^n−ν_s are K-linear combinations of powers of η^pn−ν, hence are inH.

Now by Proposition 10, 1 decomposes into a sum of pairwise orthogonal G-invariant idempotents:

1 =

n−ν−2

X

=0

p^ν−1

X

s=1,(s,p)=1

e^n−ν−_sp +

p^ν−1

X

s=0

e¹_spn−ν−1. We also have

h=

n−ν−2

X

=0

p^ν−1

X

s=1,(s,p)=1

g_1,sp+

p^ν−1

X

s=0

g_1,sp^n−ν−1. We show:

Proposition 13. For all pairs(s, )withscoprime topand0≤≤n−ν−2 or with =n−ν−1, we have

he^n−ν−_sp =g1,sp. Proof. For each pair (t, f) we have

he^n−ν−f_tpf =

n−ν−2

X

=0

p^ν−1

X

s=1,(s,p)=1

g_1,spe^n−ν−f_tpf +

p^ν−1

X

s=0

g_1,sp^n−ν−1e^n−ν−f_tpf . We have four cases to show.

Case 1. Ifn−ν−f ≥2 andt is coprime top, then g_1,sp^n−ν−1e^n−ν−f_tpf = 0.

Case 2.

g_1,sp^n−ν−1e¹_tpn−ν−1 = 0 ift6=s, while

g_1,sp^n−ν−1e¹_spn−ν−1 =g_1,sp^n−ν−1. Case 3. Fors coprime top and n−ν−≥2,

g1,spe¹_tpn−ν−1 = 0 for all t.

LINDSAY N. CHILDS

Case 4. Fors,tcoprime to p and , f ≤n−ν−2, g1,spe^n−ν−f_tpf = 0 iff 6=or iff =butt6=s, while

g_1,spe^n−ν−_spf =g_1,sp. We use Corollary 9: For all k >0 and alls, t, r,

e^r_se^r+k_t = 0 if t6≡s (modp^n−r−k)

=e^r_s ift≡s (mod p^n−r−k).

Case 1: We haven−ν−f ≥2 andtis coprime to p. Now g_1,sp^n−ν−1e^n−ν−f_tpf =z^−sdpn−1ηe¹_spn−ν−1e^n−ν−f_tpf ,

and by Proposition 8, this 6= 0 if sp^n−ν−1 ≡ tp^f (modp^{n−(n−ν−f)}). But since

ordp(tp^f) =f ≤n−ν−2<ordp(sp^n−ν−1) the congruence never holds, so Case 1 is true.

Case 2: Sinceg_1,sp^n−ν−1 is a multiple ofe¹_spn−ν−1, Case 2 follows from the orthogonality of the idempotents {e¹_tpn−ν−1}.

For Cases 3 and 4 we assume s is coprime to p and n−ν −≥ 2. We write

g1,sp=

p^{n−−ν−1}−1

X

i=0

σⁱ(z^−sdp+νηe¹_sp)

=

p^{n−−ν−1}−1

X

i=0

z^−sdp+νζ^−sdp+νη^(1+dpν)ie¹_sp(1+dp^ν)⁻ⁱ

=

p^{n−−ν−1}−1

X

i=0

γie¹_sp(1+dp^ν)⁻ⁱ

whereγi is the coefficient of e¹_sp(1+dp^ν)⁻ⁱ in theith summand.

Case 3 is the case

g1,spe¹_tpn−ν−1 =

p^{n−−ν−1}−1

X

i=0

γie¹_sp(1+dp^ν)⁻ⁱe¹_tpn−ν−1,

which = 0 from Corollary 9 by essentially the same argument as in Case 1.

Case 4: Fort coprime to pand n−ν−f ≥2, we have g_1,spe^n−ν−f_tpf =

p^{n−−ν−1}−1

X

i=0

γ_ie¹_sp(1+dp^ν)⁻ⁱe^n−ν−f_tpf . The term

γ_ie¹_sp(1+dp^ν)⁻ⁱe^n−ν−f_tpf = 0 or =γ_ie¹_sp(1+dp^ν)⁻ⁱ

depending on whether or not

sp(1 +dp^ν)⁻ⁱ≡tp^f (mod p^{n−(n−ν−f)}) that is,

sp≡tp^f(1 +dp^ν)ⁱ (modp^ν+f).

Sincesandtare coprime top, this congruence can hold exactly when=f and s≡t (modp^ν), independent of i. Thus g_1,spe^n−ν−f_tpf =g_1,sp precisely when f = and t ≡ s (mod p^ν), and = 0 otherwise. The proposition

follows.

By Proposition 13,

He^n−ν−_sp ⊇K[h]e^n−ν−_sp =K[g1,sp]e^n−ν−_sp .

NowHdecomposes into a direct sum of subrings corresponding to the idem- potents arising in Proposition 10:

H=

n−ν−2

X

=0

p^ν−1

X

s=1,(s,p)=1

He^n−ν_sp ⁻+

p^ν−1

X

s=1

He¹_spn−ν−1, so

H⊇

n−ν−2

X

=0

p^ν−1

X

s=1,(s,p)=1

K[g_1,sp]e^n−ν−_sp +

p^ν−1

X

s=1

K[g_1,sp^n−ν−1]e¹_spn−ν−1, a module over

n−ν−2

X

=0

p^ν−1

X

s=1,(s,p)=1

Ke^n−ν−_sp +

p^ν−1

X

s=1

Ke¹_spn−ν−1.

We will compute the dimension of K[g1,sp]e^n−ν−_sp over Ke^n−ν−_sp for each s, . The sum of those dimensions is less than or equal to the dimension of H as a K-module. When we show that the sum of the dimensions is pⁿ, then, sinceLM^Gis known by descent to have dimensionpⁿandH ⊆LM^G, we will obtain equality.

To compute the desired dimensions, we have

Proposition 14. For r= 1, . . . , n−−ν−1 and(s, p) = 1, g_r,sp^p =gr+1,sp.

Proof. Since gr,sp is a sum of terms involving pairwise orthogonal idempo- tents, we have

g^p_r,sp =

p^{n−−r−ν}−1

X

i=0

σⁱ(z^−sdp+r+νe^r_spη^pr) and

e^r_spη^pr =ζ^spr+e^r_sp.

LINDSAY N. CHILDS

In the summation formula forg_r,sp^p we write the index of summation in base p^{n−−r−ν−1}: i=j+kp^{n−−r−ν−1}. Then

g_r,sp^p =

p^{n−−r−ν−1}−1

X

j=0

σ^j

p−1

X

k=0

σ^{kpn−−r−ν−1}(z^−sdp+r+νe^r_spζ^spr+).

Focusing on the part of the expression forg_r,sp^p involving the indexk, we have

σ^{kpn−−r−ν−1}(z^−sdp+r+ν) =ζ^−sdkpn−1z^−sdp+r+ν, and

σ^{kpn−−r−ν−1}(e^r_sp) =e^r_sp−sdkp^n−r−1.

To verify this last formula, we see that the subscript ofe^r on the left side is sp(1 +dp^ν)^{−kpn−−r−ν−1},

and since the subscriptt ofe^r_t is defined modulo p^n−r, that subscript is sp(1 +dp^ν)^{−kpn−−r−ν−1} ≡sp−spkp^{n−−r−ν−1}dp^ν

≡sp−skdp^n−r−1 (modp^n−r).

Thus

σ^{kpn−−r−ν−1}(z^−sdp+r+νe^r_spζ^spr+)

=z^−sdp+r+νe^r_sp−skdp^n−r−1ζ^{spr+−skdpn−1}. Now we observe that

η^pre^r_sp−skdp^n−r−1 =ζ^{spr+−skdpn−1}e^r_sp−skdp^n−r−1. So

z^−sdp+r+νe^r_sp−skdp^n−r−1ζ^{spr+−skdpn−1}

=z^−sdp+r+νη^pre^r_sp−skdp^n−r−1, and the sum involving kbecomes

p−1

X

k=0

z^−sdp+r+νη^pre^r_sp−skdp^n−r−1 =z^−sdp+r+νη^pr

p−1

X

k=0

e^r_sp−skdp^n−r−1

=z^−sdp+r+νη^pre^r+1_sp

by Lemma 8, sincesd is coprime top. Thus g^p_r,sp =

p^{n−−r−ν−1}−1

X

j=0

σ^j(z^−sdp+r+νη^pre^r+1_sp )

=g_r+1,sp.