Hopf Galois Structures on Degree p

New York Journal of Mathematics

New York J. Math. 2(1996) 86–102.

Hopf Galois Structures on Degree p

Cyclic Extensions of Local Fields

Lindsay N. Childs

To Alex Rosenberg on his 70th birthday

Abstract. Let Lbe a Galois extension ofK, finite field extensions ofQp,podd, with Galois group cyclic of order p². There are p distinctK-Hopf algebras Ad, d= 0, . . . , p−1, which act onLand makeLinto a Hopf Galois extension ofK. We describe these actions. LetRbe the valuation ring ofK. We describe a collection of R-Hopf ordersEvinAd, and find criteria onEv forEv to be the associated order inAd of the valuation ringSof someL. We find criteria on an extensionL/K for Sto beEv-Hopf Galois overRfor someEv, and show that ifSisEv-Hopf Galois overRfor someEv, then the associated order AdofSinA_d is Hopf, and henceS isAd-free, for alld. Finally we parametrize the extensionsL/Kwhose ramification numbers are≡ −1 (modp²) and determine the density of the parameters of those L/Kfor which the associated order ofSinKGis Hopf.

Contents

1. Hopf Galois Structures on Galois Field Extensions 87

2. Hopf Orders 92

3. Hopf Galois Structures 96

References 102

Let pbe an odd prime, and let K be a finite extension of Qp which contains a primitive p th root of unity ζ, and with valuation ring R. Let L be a Galois extension ofKwith Galois groupGand valuation ringS. Relative Galois module theory seeks to understandSas a module over the group ringRG, or more generally over the associated orderA ofS in KG,A={α∈KG|αS⊂S}. ThenA=RG and S is RG-free of rank one if and only if L/K is tamely ramified. For wildly ramified extensions, the only general criterion available is that if the associated orderAis a Hopf order over Rin KG, thenS is A-free of rank one [Ch87]. (The converse is far from true.)

Since the work of Greither and Pareigis [GP87], one knows that L/K may be a Hopf Galois extension with respect to different Hopf Galois actions on L. In

Received November 8, 1996.

Mathematics Subject Classification. 11S15, 11R33, 16W30.

Key words and phrases. Galois module, Hopf Galois extension, associated order, wildly rami- fied, Hopf order.

c

1996 State University of New York ISSN 1076-9803/96

86

fact, Byott has recently shown that for a Galois extensionL/K with groupG, the classical Hopf Galois structure is unique if and only if the orderg ofGis coprime to φ(g) (Euler’s function) [By96]. In case L is a cyclic Galois extension of K of orderpⁿ, thenL/Khas exactlypⁿ⁻¹distinct Hopf Galois structures [Ko96]. Thus whenn = 2 there arep distinct Hopf algebrasA_d,d = 0, . . . , p−1, which give a Hopf Galois structure onL/K.

The existence of different Hopf Galois structures on L/Kraises the possibility that S may have different Galois module properties with respect to one structure than another. For example, in [CM94] we found that the associated order of the valuation ring ofQ(2¹4) in one Hopf Galois structure was Hopf and the associated order in the other structure was not. N. Byott [By96b] found a cyclotomic Lubin- Tate extension of local fields which has two Hopf Galois structures: one associated order is Hopf, while the second associated orderB is not Hopf and the valuation ring is not free overB.

In this paper we describe as algebras the Hopf algebrasA_d which make L/K Hopf Galois, and their actions onL. Following [Gr92], we construct a collection of Hopf ordersE_v overR inside eachA_d. We find criteria onL/Kin order thatS be a Hopf Galois extension ofR for someE_v. This implies, by [Ch87], that E_v is the associated order ofS inA_d. In contrast to the examples just described, however, it turns out that ifSis Hopf Galois overRforE_v, a Hopf order inA_dfor somed, then the associated order ofSinA_dfor everydis Hopf, in particular forA₀=KG. Thus in the case of cyclic Galois extensions of degreep², the non-classical Hopf Galois structures on L do not “tame” the wild extension L/K better than the classical structure given by the Galois group.

We apply Greither [Gr92] to find necessary and sufficient conditions on an order E_vto be realizable: that is, to be the associated order of the valuation ring of some extension L/K: the congruence condition on v is the same as for Hopf orders in KGas found by Greither. Finally, we quantify the remark in [Gr92, Remark (c), page 63] that congruence conditions on the ramification numbers of a cyclic totally ramified extensionL/K of degreep² are “badly insufficient” for deciding whether the valuation ringS ofL is Hopf Galois overR.

The concept of Hopf Galois extension of commutative rings arose in [CS69] as a merger of M. Sweedler’s work on Hopf algebras and the development of Galois theory of commutative rings by S. U. Chase, D. K. Harrison and Alex Rosenberg [CHR65].

1. Hopf Galois Structures on Galois Field Extensions

We begin by recalling the main result of Greither and Pareigis [GP87].

Greither-Pareigis. IfLis a Galois extension ofK with groupG, then there is a bijection between Hopf Galois structures onL/Kand regular subgroups ofP erm(G) normalized byλ(G).

HereP erm(G) is the group of permutations of the setG, λ(G) is the image of Gin P erm(G) given by left translation, and a subgroupN of P erm(G) is regular ifN acts transitively, has order equal to the order ofG, and the stabilizer in N of any element ofGis trivial. (Any two of these last conditions implies the third.)

IfN is a regular subgroup ofP erm(G), then the group ringLN acts onGL:=

M ap(G, L) byaη(f)(σ) =af(η⁻¹(σ)) forainL,σinG,f inGL,η inN. Thus if

e_σ is the function which sendsσto 1 andτ to 0 ifτ=σinG, andη is inN, then η(e_σ) =e_η(σ). This yields a map

LN×GL→GL.

The Hopf Galois structure onL is obtained by taking the fixed rings of LN and GLunder the action ofG, whereGacts onGLbyσ(ae_τ) =σ(a)e_στ, and acts on LN byσ(aη) =σ(a)σ(η): the action ofσinGonη inN is by conjugation byλ(σ) inP erm(G).

LetGbe cyclic of orderpⁿ. Then Kohl [Ko96] has shown that the only regular subgroups N of P erm(G) normalized by λ(G) are isomorphic to G, and hence (cf. also [By96, Lemma 1, (i)]) there are exactlypⁿ⁻¹ suchN.

We restrict to the casen= 2. Then we have

Proposition 1.1. The subgroups ofP erm(G)normalized byλ(G)areN_d ford= 0,1, . . . , p−1, whereN_d=ηwith η(σⁱ) =σ(i−1)(1+pd).

These groups were found by using [By96, Proposition 1], a refinement of [Ch89, Proposition 1].

Proof. Clearlyη is inP erm(G). One verifies by induction that for anyr, η^r(σⁱ) =σ^{(i−r)+(ir−r(r+1)2 )pd}.

Henceη has orderp² and the stabilizer inN_d of anyσⁱ is trivial. SoN_d is regular.

Also, for anyd, N_d⊂P erm(G) is normalized byλ(G). In fact, λ(σ)ηλ(σ⁻¹) =η^1+pd.

For

λ(σ)ηλ(σ⁻¹)(σⁱ) =λ(σ)η(σⁱ⁻¹)

=λ(σ)(σ(i−2)(1+pd))

=σ(i−1)+(i−2)pd, while

η^1+pd(σⁱ) =σi−(1+pd)+(i−1)pd

=σ(i−1)+(i−2)pd.

Example 1.2. Forp= 3, setd= 1, then η is the permutation which sendsσⁱ to σ⁴⁽ⁱ⁻¹⁾; its cycle representation is

(0,5,7,6,2,4,3,8,1).

We have an actionLN×GL→GL, which we will describe below. Looking at the fixed elements under the action ofG, we have, first, that

(GL)^G=

τ

a_τe_τ:

a_τe_τ=

σ(a_τ)e_τ

=

τ

a_τe_τ:a_στ =σ(a_τ)

=

σ

σ(a)e_σ

This is isomorphic toLunder the map sendingain Lto σ(a)e_σ. Now identifyσin Gwithλ(σ) inP erm(G). Then,

LN^G=

a_iηⁱ:

a_iηⁱ=

σ(a_i)σ(ηⁱ)

where σ(ηⁱ) means the elementη₀ of N so thatη₀ =λ(σ)ηⁱλ(σ)⁻¹ in P erm(G).

Now

σ(η) =σησ⁻¹=η^1+pd

as we observed above, and henceσ(ηⁱ) = η^i(1+dp), and soσ^k(ηⁱ) = η^i(1+kdp). In particular,η^p is fixed under the action ofG.

LetN^p=η^pand let

e_s= (1/p)

p−1

i=0

ζ^−siη^pi

in KN^p. The e_s for s = 0, . . . , p−1 are the pairwise orthogonal idempotents of KN^pcorresponding to the distinct irreducible representations ofKN^p: η^pe_s=ζ^se_s for alls.

For v in L, set a_v = _p−1

s=0v^se_s. These elements, defined by Greither [Gr92], are the elements ofLN^p corresponding to the tuple (1, v, v², . . . , v^p−1) under the isomorphism betweenLN^pandL×L×· · ·×Linduced byη^p→(1, ζ, ζ², . . . , ζ^p−1).

Thusa_vw =a_va_w for allv, winL.

Proposition 1.3. Let L^σp = M = K[z] wherez^p is in K and σ(z) = ζz. Let LN^G correspond to the embedding β of G into Hol(N)so that β(σ) =ηγ where γηγ⁻¹=η^1+pd. ThenLN^G=K[η^p, a_vη]wherev=z^−d.

Proof. We have that σ^k(η) = η^1+kpd, so σ^p(η) = η^1+p2d = η. So σ^p fixes the elements ofN, andLN^G=M N^G. SinceGfixesη^p and

e_s= (1/p)

p−1

i=0

ζ^−siη^pi,

Gfixes the idempotents e_sfor alls. Hence

σ(a_z−dη) =η^1+pd

p−1

s=0

σ(z^−ds)e_s

=η

p−1

s=0

ζ^−dsz^−dsη^pde_s

=η

p−1

s=0

ζ^−dsz^−dsζ^dse_s

=η

p−1

s=0

z^−dse_s

=a_z−dη.

ThusK[η^p, a_vη]⊂LN^G. But by Galois descent,LN^Ghas rank p² overK, and sincea_vp is in K[η^p], one easily sees that (a_vη)^p is in K[η^p], henceK[η^p, a_vη] has

rankp²overK, hence equality.

We observe for later use that K[η^p, a_vη] = K[η^p, a_vcη] for any c in K. For a_vc=a_va_c, soa_vcη=a_c·a_vη, and a_c is inK[η^p].

LetA_d denote theK-Hopf algebra K[η^p, a_vη] with v =z^−d. We examine the action ofA_d=LN^GonL.

Since L/K is a Galois extension with Galois group G = C_p2 = σ and K containsζ, a primitive pth root of unity, we can assume that M =L^σp =K[z]

with z^p in K and σ(z) = ζz, and L =M[x] with x^p in M andσ^p(x) =ζx. Let v=cz^−d,withcin K and 0≤d≤p−1.

Proposition 1.4. A_d=K[η^p, a_vη] acts onL=K[z][x]by η^p=σ^p

and for ainK[z]

(a_vη)(ax^m) =v^mσ(ax^m).

In particular,A₀=K[η] with η(s) =σ(s) forsin L, the classical action by the group ring of the Galois groupG.

Proof. We identifyLas a subset ofGL=M ap(G, L) via the isomorphism

a→

p−1

i=0

σⁱ(a)e_i

wheree_i=e_σi. Then as we observed in the proof of Proposition 1.1, η^r(e_i) =e_{i−r−pd(ir−}r(r+1)

2 ).

In particular,η^pk(e_i) =e_i−pk, so η^p

σⁱ(a)e_i =

σⁱ(a)e_i−p

=

σ^i+p(a)e_i

=

σⁱ(σ^p(a))e_i which corresponds toσ^p(a) in L.

Now forain K[z],

(a_vη)(ax^m) =

⎛

⎝

s,k

1

pv^sζ^−ksη^kp+1

⎞

⎠(ax^m)

=

s,k

1

pv^sζ^−ksη^kp+1

i

σⁱ(ax^m)e_i

=

i,s,k

1

pv^sζ^−ksσⁱ(ax^m)e(i−kp−1)+pd(i−1). The subscript oneis modp², so if we set

j=i(1 +pd)−(1 +kp+dp), then

i≡j(1−pd) + (1 +kp) (modp²)

= (j+ 1) +p(k−jd) and the sum becomes

=

j,s,k

1

pv^sζ^−ksσ(j+1)+p(k−jd)(ax^m)e_j. Sinceσ^p fixesain M =K[z], this is

=

j,s,k

1

pv^sζ^−ksσ^j+1(ax^m)ζ^(k−jd)me_j

=

j

s

v^s

1 p

k

ζ^−ks+km

σ^j+1(ax^m)ζ^−jdme_j.

The sum overkispifs=m and 0 otherwise. So the sum overj andsbecomes

=

j

v^mζ^−jdmσ^j+1(ax^m)e_j.

Nowv=cz^−d, so

σ^j(v^m) =c^mζ^−jdm(z^−dm)

=ζ^−jdmv^m. Thus the sum

=

j

σ^j(v^m)σ^j+1(ax^m)e_j

=

j

σ^j(v^mσ(ax^m))e_j

which corresponds tov^mσ(ax^m) inL. That is, (a_vη)(ax^m) =v^mσ(ax^m).

2. Hopf Orders

Now supposeKis a finite extension ofQp, with valuation ringRand parameter π. Lete be the absolute ramification index ofK. AssumeK contains a primitive pth root of unityζ. Then (ζ−1)R=π^eRand (p−1)e=e.

LetM =K[z] withz^p=binR, and letT be the valuation ring ofM. Then we may consider theK-Hopf algebrasA_d =K[η^p, a_vη], wherev =z^−d, as described in Section 1. (Recall that for anycinK,K[η^p, a_vη] =K[η^p, a_vcη]). In this section we extend work of Greither [Gr92][GC96] to construct a collection of Hopf orders overR in A_d for eachdwith 0≤d≤p−1. These Hopf orders are parametrized by integersi, j with 0≤i, j≤e and a unitcin R.

Forian integer, 0≤i≤e, leti =e−i.

Theorem 2.1. Let i, j be integers with0 < i, j ≤e. Let H_i =R η^p−1

πⁱ

, a Hopf order inK[η^p]. Forv=z^−dc, cinR, lety=^av_π^η−1_j . Then theR-algebraE=H_i[y]

is an R-Hopf order in A_d =K[η^p, a_vη] and a Hopf algebra extension of H_j by H_i if and only if

ζb^−dc^p≡1 (modπ^i+pjR) and

b^−dc^p≡1 (modπ^pi+jR).

Recall that theH_ifor 0≤i≤eare all the Hopf orders in the group ringK[η^p] by Tate-Oort [TO70]. This description of theH_i goes back to Larson [La76].

Proof. The canonical map fromK[N] toK[N/N^p] sendsη^p to 1, and sendsa_vto 1 andH_itoR, so the image ofEisR[^η−1¯

π^j ] =H_j. To show thatEis a Hopf algebra extension ofH_j byH_i, we need to show thatE∩K[η^p] =H_i. This is equivalent to showing that the monic polynomial of degree p satisfied by y over K[η^p] has coefficients inH_i. We follow [GC96, Section 2] and utilize [Gr92, I, section 3].

Nowa_vη= 1 +π^jy, so

(a_vη)^p= (1 +π^jy)^p

= 1 +

p−1

r=1

p r

π^jry^r+π^jpy^p,

hence

y^p+π^−jp

p−1

r=1

p r

π^jry^r+1−(a_vη)^p π^jp = 0.

Note that (a_vη)^p=a_vpη^p, andη^p=a_ζ, so (a_vη)^p =a_vpζ. Thusy satisfies a monic polynomial with coefficients inH_i if and only if inH_i,

1) π^jpdivides pπ^jr forr= 1, . . . , p−1;

2) π^jpdivides 1−a_vpζ.

Condition 1) is equivalent tojp≤e+j, orj≤e. Condition 2) is the same as

a_vpζ ≡1 (modπ^jpH_i), which, by [Gr92, I 3.2b], is equivalent to

v^pζ≡1 (modπ^i+pjR), or, sincev^p =b^−dc^p,

b^−dc^pζ≡1 (modπ^i+pjR).

Note that ifj≤ethen ^1−(a_πpj^vη)p ∈E∩K[η^p], so if ^1−(a_πpj^vη)p ∈/ H_ithenE∩K[η^p]= H_i.

Now we show that E is closed under comultiplication if and only if v^p ≡ 1 (mod π^pi+jR).

Recall thatA_d=K[η^p, a_vη] andT is the valuation ring ofM. LetE=R[t][y] = H_i[y] with t=^ηp−1

πⁱ , y= ^avη−1

π^j . Since Δ is an algebra homomorphism, to show E is a coalgebra, it suffices to show that Δ(y)∈E⊗E.

Now Δ(y)∈A_d⊗A_d=K⊗R(E⊗RE) andR is integrally closed. If we show that Δ(y)∈T⊗R(E⊗RE) =T E⊗TT E, then, sinceE and thereforeE⊗RE are freeR-modules,

(T⊗R(E⊗RE))∩(K⊗R(E⊗RE)) =E⊗RE, and so Δ(y)∈E⊗E.

We will show, in fact, that

Δ(y)∈C⊗C

whereC=H_i·1 +H_i·y. Again, it is enough to show that Δ(y)∈T C⊗_T T C.

Now

Δ(y) = Δ

a_vη−1 π^j

= Δ(α_vη)−a_vη⊗a_vη

π^j +y⊗(1 +π^jy) + 1⊗y and the last two terms are inC⊗C. So it suffices to show that

Δ(a_vη)−a_vη⊗a_vη

π^j ∈T C⊗TT C.

Now a_v is a unit of T H_i. For since v^p ∈ U_pi+j(R), then v ∈ U_pi+j(T), hence by [Gr92, I 3.2(b)], a_v ∈ 1 +π^j/pH_i. Since j > 0, a_v is a unit of T H_i. Since a_vη= 1 +π^jt∈T H_i·1 +T H_i·t=T C, thereforeη∈T C. So

Δ(a_v)−a_v⊗a_v π^j

(η⊗η)∈T C⊗TT C if and only if

Δ(a_v)−a_v⊗a_v

π^j ∈T H_i⊗T T H_i. To decide if

Δ(a_v)−a_v⊗a_v

π^j ∈T H_i⊗TT H_i

we identify elements ofM[η^p]⊗M M[η^p] as p×pmatrices as in [Gr92, I, Section 3].

We have

Δ(a_v)−a_v⊗a_v

π^j = 1

π^j

p−1

s=0

⎡

⎣Δ(v^se_s)−

0≤r,t<p,r+t≡s(modp)

v^re_r⊗v^te_t

⎤

⎦

=

p−1

s=1

v^s

r+t≥p,r+t≡s(modp)

1−v^p π^j e_r⊗e_t

.

Let ^1−v_πj^p =w. Then

Δ(a_v)−a_v⊗a_v π^j

corresponds to the matrix M = {M_a,b} where M_a,b is the coefficient of e_a ⊗e_b. Here,M_a,b= 0 ifa+b < p, andM_a,b =wv^s wherea+b=p+sfora+b≥p.

Now ^Δ(av)−a_πj^v⊗av ∈T H_i⊗T H_i is equivalent, by [Gr92, I, Lemma 3.3] to: for allk, k^∗ with 0≤k, k^∗< p,π^i(k+k∗) divides

d^k,k∗(M) = k a=0

k^∗

b=0

k a

k^∗ b

(−1)^a+bM_a,b

= l s=0

a+b=p+s

k a

k^∗ b

(−1)^a+bM_a,b

wherek+k^∗=p+l. SinceM_a,b=wv^s fora+b=p+s, this is

=w l s=0

a+b=p+s

k a

k^∗ b

(−1)^p+sv^s

=w l s=0

k+k^∗ p+s

(−1)^p+sv^s. Now sinces < p,

k+k^∗ p+s

= p+l

p+s

≡ l

s

(mod p), so

≡w l s=0

l s

(−1)^p+sv^s (mod p)

≡ −w(1−v)^l (mod p).

ThusM ∈T H_i⊗T H_i if and only ifπ^i(k+k∗) =π^i(p+l) divides w(1−v)^l for all l≥0.

For l = 0 the condition is: π^ip divides w = ^1−v_πj^p, or v^p ≡ 1 (modπ^pi+j).

Assumingv^p≡1 (modπ^pi+j), then, sincev∈U_pi+j(T), v−1∈π^i+pjT

(recall: πis the parameter forR), so

(v−1)^l∈π^il+jlpT . Alsow∈π^piR, so

w(1−v)^l∈π^pi+il+jlpT .

Sincei(k+k^∗) =pi+il, thereforeπ^i(k+k∗)divides d^k+k∗(M) for allk, k^∗. Thus

Δ(a_v)−a_v⊗a_v

π^j ∈T H_i⊗T H_i

if and only ifv^p≡1 (modπ^pi+j). That completes the proof.

Supposei, j satisfy 0< i, j≤e and consider the two conditions v^p≡1 (modπ^pi+j);

ζv^p≡1 (modπ^i+pj).

Since

ζv^p−1 =ζv^p−v^p+v^p−1

= (ζ−1)v^p+ (v^p−1)

we must have two oford_R(ζv^p−1),ord_R(v^p−1) andeequal, and both≤the third (isosceles triangle inequality). ForE to be a Hopf algebra and a free H_i-module requires

ord_R(ζv^p−1)≥i+pj and

ord_R(v^p−1)≥pi+j.

Thusi+pj≤e or pi+j ≤e. The first is equivalent to i ≥pj; the second to j≥pi. Hence:

Corollary 2.2. In order thatE be a Hopf algebra,iandj must satisfy: 0< i, j≤

e andi≥pj or j≥pi.

Note: i≥pjis the condition of [Gr92, I 3.6] and [Gr92, II], cf. [Un94].

Ifi+j≤e, theni+pj≤pi+j, so if ord_R(v^p−1)≥pi+j, then ord_R(ζv^p−1)≥min{e, ord_R(v^p−1)}

≥min{e, pi+j} ≥i+pj.

So we have

Corollary 2.3. If i, j > 0, i+j ≤ e and i ≥ pj, then E is a Hopf order with E∩K[η^p] =H_i if and only iford_R(v^p−1)≥pi+j.

The Hopf algebrasEpresumably fit within the classification of [By93], but the description of theE here is rather different that that of Byott.

3. Hopf Galois Structures

Now we consider a cyclic extensionL/Kwith Galois groupG=σof orderp², and see whenS/RisE_v-Galois for somev.

We assume throughout this section that i, j > 0, 0 ≤ i+j ≤ e and i ≥ pj.

Under these hypotheses,p(i+j)≤pj+ 1. For sincepj≤i, we have pi≥p²j >2pj−1

so

1−pj >−pi+pj, 1 +pe−pj > pe−pi+pj, which is

pj+ 1> p(i+j).

SupposeS/Ris E_v-Galois. Then T /R isH_j-Galois andS/T is T⊗H_i-Galois, by [Gr92]. Sincei, j >0,M/KandL/M are totally, hence wildly ramified.

If T /R is H_j-Galois, then (cf. [Ch87]) M = K[z] with z^p = 1 +uπ^pj+1 and t= ^z−1

π^j is a parameter forT, soT =R[t]. Sinceσ(t) = ^ζ−1

π^j z+t=t+ut^pj foru some unit ofT, the ramification numbert^G/H₁ =pj−1. The converse also holds:

c.f [Ch87] or [Gr92]. By [Se62, Ch. V, Sec. 1, Cor. to Prop. 3],t^G/H₁ = t^G₁, so t^G₁ =pj−1.

Similarly, ifS/T isT⊗H_i-Galois,M/Kis totally ramified, andtis a parameter for T, we may find x in L so that L = M[x] with σ^p(x) = ζx and x^p = γ = 1 +ut^p2i+1 for some unituofT. Thenw=^x−1

πⁱ is a parameter forS, and σ^p(w) =ζ−1

πⁱ x+w=w+w^p2iu

for some unit u of S. So the ramification number for L/M ist^H₁ =p²i−1, and conversely. Sincet^H₁ =t^G₂, we havet^G₂ =p²i−1.

Now L is a Galois extension of K with group G = σ, cyclic of order p², so σ(x) = βxfor some β in T with N_M/K(β) = ζ.If ord_T(x^p−1) = p²i+ 1, then σ(w) = ^β−1

πⁱ x+w, so since t^G₁ =pj−1,ord_L(^β−1

πⁱ ) =pj. Thus ord_L(β−1) =p²i+pj

and so

ord_M(β^p−1) =p²i+pj.

Lemma 3.1. β is unique modulot^pi+pjT.

Proof. Letγ=x^p= 1 +ut^p2i+1 for some unituofT. Suppose we replacexbyxαfor some α∈T. Then

(xα)^p =γα^p= (1 +ut^p2i+1)α^p.

Iford_T((xα)^p−1) = p²i+ 1, thenord_T(α^p−1) ≥p²i+ 1. If ord_T(α−1) = s, thenord_T(α^p−1) =psunlesspe ≤s. Assuming s≤pe, then we require

ps≥p²i+ 1, so

s≥pi+ 1.

Now if we replacexbyxα, thenσ(xα) =β^σ(α)_α (xα), soβ is replaced byβ^σ(α)_α . If ord_T(α−1) =sthen by [Wy69, Theorem 22],

ord_T σ(α)

α −1

≥s+pj−1

≥pi+ 1 +pj−1 =p(i+j).

Soβ^σ(α)_α ≡β (mod t^p(i+j)T).

Thusβ is unique modulot^p(i+j)T.

Given L/K with ramification numbers t^G₁ =pj−1 and t^G₂ = p²i−1, when is there someE_v so thatS/RisE_v-Galois? Since the discriminant overRofSequals the discriminant of the dual ofE_v, Swill beE_v-Galois if and only ifE_v acts onS (see [Gr92, II, Section 1]), that is,ξ·s is in S (not just in L) for all ξ∈ E_v and s∈S. Equivalently,E_v⊂ A, the associated order ofS in A_d.

We knowAis an algebra. So to showE_v⊂ Ait suffices to show that t= η^p−1

πⁱ ∈ A and

y= a_vη−1 π^j ∈ A. Now

Δ(t) = η^p⊗η^p−1⊗1 πⁱ

=

η^p−1 πⁱ

⊗η^p+ 1⊗

η^p−1 πⁱ

=t⊗(1 +πⁱt) + 1⊗t.

Hence if

t z−1

π^j

∈S, then sinceL is anA_d-module algebra,

t

R z−1

π^j

⊂S,

sotT ⊂S. Also, if

t x−1

πⁱ

∈S then

t

T x−1

πⁱ

⊂S, sotS⊂S andt∈ A. HenceH_i⊂ A.

Similarly, we showed in the proof of Theorem 2.1 thatC =H_i·1 +H_i·y is a subcoalgebra ofE_v. If

y z−1

π^j

∈S then

C z−1

π^j

⊂S,

soCT ⊂S. Also, if

y x−1

πⁱ

∈S then

C x−1

πⁱ

⊂S, so, since

S=R z−1

π^j

x−1 πⁱ

,

CS⊂S. SoC⊂ A. SinceC generatesE_v as anR-algebra,E_v⊂ A. ThusE_v acts onS if and only ift=^ηp−1

πⁱ andy=^avη−1

π^j map ^z−1

π^j and ^x−1

πⁱ into S.

We see that

t z−1

π^j

= 0, y

z−1 π^j

= σ⁻¹(z)−z

π^e =ζ⁻¹−1 π^e z∈T , and

t x−1

πⁱ

=ζ⁻¹−1 π^e x∈S;

finally, by Proposition 1.4, y

x−1 πⁱ

= a_vη(x)−x

π^i+j =vσ(x)−x

π^i+j =vβ−1 π^i+j x is inS if and only if

β ≡v⁻¹ (mod π^i+jT).

From this we have

Proposition 3.2. Let L/K be a Galois extension with group G cyclic of order p² and with ramification numbers t₁ =pj−1 and t₂ =p²i−1, where i, j satisfy the inequalities at the beginning of this section. Then the valuation ring S of L is E_v-Hopf Galois over R, and hence the associated order of S inA_d is Hopf, if and

only ifβ≡v⁻¹ (mod π^i+jT).

Now we observe

Lemma 3.3. If v≡z^−dc for somec inR, thenv≡c (modπ^i+jT).

Proof. We have

z= 1 +ut^pj+1, ua unit ofT. Sincepj+ 1> p(i+j),

z≡1 (mod π^i+jT =t^p(i+j)T).

Corollary 3.4. With the hypotheses of Proposition 3.2, if S is E_v-Galois then p dividesj.

Proof. We haveord_T(β−1) =pi+j, and soord_T(v⁻¹−1) =ord_T(v−1) =pi+j.

Henceord_R(v^p−1) =pi+j.

Sincev=z^−dc andpi+j < pj+ 1, we have

ord_R(v^p−1) =pi+j < pj+ 1 =ord_R(z^p−1),

soord_R(v^p−1) =ord_R(c^p−1) =p ord_R(c−1). Henceord_R(c−1) =i+j/p, and

pdivides j.

Corollary 3.5. With the hypotheses of Proposition 3.2, ifS/R is Hopf Galois for someE_v, thenS is free over the associated order inA_d for alld.

Proof. We have thatS/Ris Hopf Galois for E_v,v=z^−dc, if and only if β ≡(z^−dc)⁻¹ (mod π^i+jT).

But

z^−d≡1 (mod π^i+jT), and hence

β ≡(z^−dc)⁻¹ (mod π^i+jT)

for everyd, and soE_vacts onSwhenv=z^−dcfor everyd. Hence for anyd,S/Ris E_z−dc-Hopf Galois, and soE_z−dc is the associated order ofS inA_d for everyd.

Corollary 3.6. E_v is realizable if and only iford_T(v−1) =pi+j.

Proof. IfL/KrealizesE_v, that is,E_v is the associated order of the valuation ring of the Galois extension L of K, then, as we showed, β ≡ v⁻¹ (modπ^i+jT), so ord_T(v−1) =pi+j. Conversely, iford_T(v−1) =pi+j, then sincev=cz^−d for somec∈R,ord_T(c−1) =pi+j, soE_c is realizable by someL/Kby [Gr92, Part II, Section 3]. But then, sincecz^−d ≡c (modπ^i+jT), we see that the extension

L/Kalso realizesE_v by Proposition 3.2.

The problem raised at the beginning of this section can be precisely answered by the following corollary, in which the hypotheses onLare recapitulated.

Corollary 3.7. Let K be a finite extension of Qp containing ζ_p, a primitive pth root of unity. Let L be a cyclic Galois extension of K with Galois group G=σ of degreep² with intermediate field M and with ramification numberst^G₁ =pj−1 andt^G₂ =p²i−1 where0< pj≤i, pdividesj, and i+j≤e =e_K/Qp/(p−1). Let S, T andR be the valuation rings ofL, M andK, respectively. LetL=M[x] with ord_M(x^p−1) =p²i+ 1 andσ(x) =βx. Then S is an E_v-Hopf Galois extension ofR if and only if β is congruent to an element ofR modulo t^pi+pjT =π^i+jT .