de Bordeaux 19(2007), 337–355

## A classification of the extensions of degree p

^{2}

## over Q

^{p}

## whose normal closure is a p-extension

parLuca CAPUTO

R´esum´e. Soitk une extension finie de Qp et soit Ek l’ensemble
des extensions de degr´ep^{2} surk dont la clˆoture normale est une
p-extension. Pour chaque discriminant fix´e, nous calculons le nom-
bre d’´el´ements deE_{Q}_{p}qui ont un tel discriminant, et nous donnons
les discriminants et les groupes de Galois (avec leur filtrations des
groupes de ramification) de leurs clˆotures normales. Nous mon-
trons aussi que l’on peut g´en´eraliser cette m´ethode pour obtenir
une classification des extensions qui appartiennent `a Ek.

Abstract. Let k be a finite extension of Qp and Ek be the set
of the extensions of degree p^{2} over k whose normal closure is a
p-extension. For a fixed discriminant, we show how many exten-
sions there are in E_{Q}_{p} with such discriminant, and we give the
discriminant and the Galois group (together with its filtration of
the ramification groups) of their normal closure. We show how this
method can be generalized to get a classification of the extensions
inEk.

1. Notation, preliminaries and results.

Throughout this paper, p is an odd prime and k will be a fixed p-adic field of degreedoverQp which does not contain any primitive p-th root of unity. IfE is ap-adic field andL|E is a finite extension, then we say that L|E is ap-extension if it is Galois and its degree is a power of p.

The aim of the present paper is to give a classification of the extensions of
degreep^{2} overQpwhose normal closure is ap-extension. This classification
is based on the discriminant of the extension and on the Galois group and
the discriminant of its normal closure. Let E_{k} be the set of the extensions
of degree p^{2} over kwhose normal closure is a p-extension. Then for every
L∈ E_{k}, there exists a cyclic extension K|kof degree p,K ⊆L and L|K is
cyclic (of degree p). Furthermore, the converse is true: if K|k is a cyclic
extension of degree p, then every cyclic extension L of degree p over K
is an extension of degree p^{2} over k whose normal closure is a p-extension

Manuscrit re¸cu le 6 juin 2005.

(see Prop. 2.1). Therefore, if K|k is a cyclic extension of degreep, we can
consider the subset E_{k}(K) of E_{k} made up by the extensions in E_{k} which
containK. The idea is to study the compositumM_{k}(K) of the extensions
inE_{k}(K). ClearlyM_{k}(K) is the maximal abelianp-elementary extension of
K: it is easy to prove thatMk(K)|kis Galois (see Prop. 2.2). We describe
the structure of the Galois groupG_{k}(K) =Gal(M_{k}(K)|k) (see Prop. 3.1),
using results both from classical group extensions theory (see [5]) and from
[3]. G_{k}(K) is a p-group of order p^{dp+2} which admits a presentation with
d+ 1 generators.

Then we focus on the case k = Qp. We put G_{p}(K) = G_{Q}_{p}(K). Once
one has a description of the normal subgroups of Gp(K) (see Prop. 4.1),
it is not difficult to describe the quotients ofG_{p}(K) (see Lemma 4.1). We
are able as well to decide if a quotient ofGp(K) is the Galois group of the
normal closure of an extension in E_{Q}_{p}(K) and, if this is the case, it is easy
to give the number of extensions whose normal closure has that group as
Galois group (see Section 5).

Finally using class field theory and [2], we determine the ramification
groups of Gp(K) (we distinguish the case when K is unramified from the
case when K is totally ramified, see respectively Section 6 and Section
7). This allows us to determine, after some standard computations, the
possible values for the discriminant of the extensions inE_{Q}_{p}(K) as well as
the possible values for the discriminant of their normal closures. We collect
the results in a table (see Section 8).

It would not be difficult to generalize the results of Sections 4-7 to an
arbitrary groundp-adic fieldk. Then the method used in the present paper
could be generalized to give a classification, for example, of the extensions
of degree p^{3} over Qp whose normal closure is a p-extension. In fact, if L
is one of these extensions, then there exists a cyclic extension K|Qp such
that K ⊆L and L|K is an extension of degree p^{2} whose normal closure a
p-extension (see Prop. 2.1).

Acknowledgements. The results presented here come from my master thesis which was made at the University of Pisa under the direction of Prof. Roberto Dvornicich. I would like to express my thanks to him for his supervision and his advice.

2. Some properties of the extensions of degreep^{2} overk whose
normal closure is a p-extension.

Proposition 2.1. LetL|k be an extension of degreep^{s},s∈N. We denote
by Lthe normal closure of Lover k. ThenL|kis a p-extension if and only
if there exists a tower of extensions of k, say

k=L^{(0)} ⊆L^{(1)} ⊆. . .⊆L^{(s−1)}⊆L^{(s)}=L,

such that, for eachi= 0, . . . , s−1, L^{(i+1)}|L^{(i)} is cyclic of degree p.

Proof. We proceed by induction on s. Suppose s = 1, i.e. [L : k] =
p. Assume first [L : k] = p^{n}: then Gal(L|L) is a maximal subgroup of
Gal(L|k). In particularGal(L|L) is normal in Gal(L|k). Therefore L=L
and we get what we want (L|kis cyclic of degreep). The other implication
is obvious.

Now suppose that the proposition is true for s∈ N. Assume first that
[L : k] =p^{s+1}. If [L : k] = p^{n}, there must exist H < Gal(L|k) maximal
(and therefore normal) such that Gal(L|L) ⊆ H. Put k^{0} = F ixH: then
k^{0}|kis cyclic of degreep. Then [L:k^{0}] =p^{s} and, by induction, there exists

k^{0} =L^{(1)}⊆L^{(2)}⊆. . .⊆L^{(s−1)} ⊆L^{(s)}=L
as in the claim. But then

k=L^{(0)} ⊆k^{0} =L^{(1)} ⊆. . .⊆L^{(s−1)}⊆L^{(s)} =L

is the sequence we are looking for. Conversely, assume that there exists a
sequence with the properties of the claim, putk^{0} =L^{(1)} and let L^{0} be the
normal closure ofLoverk^{0}. By induction, L^{0} is a p-extension ofk^{0}. IfL^{0}|k
is normal, there is nothing to show. Otherwise the normalizer ofGal(L|L^{0})
in Gal(L|k) must be Gal(L|k^{0}) and the latter is normal in Gal(L|k). L
is the compositum of the conjugates of L^{0} over k: each of them contains
k^{0} and is a p-extension of k^{0}. The compositum of p-extensions is again a
p-extension and then Lis a p-extension.

Remark. We recover the well known result which says that an extension
of degree p, whose normal closure is a p-extension, is cyclic (the highest
power ofp which divides p! is p). We shall use Prop. 2.1 for s= 2: then
for an extensionLof degree p^{2} overkthe following are equivalent:

• the normal closure Lof Loverk is ap-extension;

• there exists a cyclic extensionK of degreepoverksuch thatK ⊆L and L|K is Galois.

We define now some notation which will be used in what follows. LetE_{k}
be the set of the extensions of degree p^{2} overk whose normal closure is a
p-extension. For any cyclic extension K of degree p of k, we defineE_{k}(K)
to be the set of the estensions inE_{k} which containK. Then it is easily seen
thatE_{k}=SE_{k}(K), the union being taken over the set of cyclic extensions
K of degreep overk. Moreover E_{k}(K) is the set of the cyclic extensions of
K of degree p.

Proposition 2.2. Let K be a cyclic extension of degree p over k. Then
there exists one and only one extension M_{k}(K) of k such that

(i) K ⊆M_{k}(K),

(ii) M_{k}(K) is Galois over K and Gal(M_{k}(K)|K)∼= (Z/pZ)^{pd+1}.

Moreover, M_{k}(K) is Galois over k and, if K^{0} is another Galois extension
of degree p over k, we have

Gal(Mk(K)|k)∼=Gal(Mk(K^{0})|k).

Proof. Since K has degree p over k (in particular it does not contain any
primitivep-th root of unity), we know that K^{∗}/K^{∗p} ∼= (Z/pZ)^{pd+1}. Then
we letM_{k}(K) be the extension of K which corresponds by local class field
theory toK^{∗p}: M_{k}(K) is the compositum of the cyclic extensions of degree
p overK and Gal(M_{k}(K)|K)∼= (Z/pZ)^{pd+1}. In particular M_{k}(K) verifies
(i) and (ii) and it is clearly unique.

Now, let Qp be an algebraic closure and we considerMk(K) ⊆Qp. Let
σ : Mk(K) → Qp be an embedding over k. Since K is normal, we have
σ(K) = K then σ(M_{k}(K)) is an extension of degree p^{pd+1} of K. In fact
it is Galois over K: for, if τ :σ(M_{k}(K))→ Qp is an embedding over K,
denoting again withσ any extension ofσ, we haveσ^{−1}τ σ|_{K} = id_{K}. Using
the normality of Mk(K) over K, we obtain τ σ(Mk(K)) = σ(Mk(K)), i.e.

σ(M_{k}(K)) is Galois overK. At the same time, we obtain an isomorphism
between the Galois groups ofσ(M_{k}(K)) andM_{k}(K) overK (which isτ 7→

σ^{−1}τ σ); from the uniqueness,σ(M_{k}(K)) =M_{k}(K).

Let K^{0} be an other Galois extension of degree p over k. We denote
by ρ the restriction homomorphism from Gal(Mk(K^{0})|k) to Gal(K^{0}|k) ∼=
Z/pZ. Using [3], we see that there exists a Galois extension M^{0} over k
with Galois group isomorphic to Gal(Mk(K^{0})|k) which contains K and
such that the restriction ρ^{0} from Gal(M^{0}|k) to Gal(K|k) coincides with ρ:

then kerρ∼= kerρ^{0}. From this it follows thatM^{0} is a Galois extension ofK
such thatGal(M^{0}|K)∼= (Z/pZ)^{pd+1} and then M^{0} =M_{k}(K). In particular,
Gal(M_{k}(K^{0})|k)∼=Gal(M^{0}|k) =Gal(M_{k}(K)|k).

We will denoteGal(Mk(K)|k) byGk(K). Furthermore we put

M_{Q}_{p}(K) =Mp(K), G_{Q}_{p}(K) =Gp(K), E_{Q}_{p} =E_{p}, E_{Q}_{p}(K) =E_{p}(K).

Remark. It is clear that the compositum of the extensions belonging to
E_{k}(K) is equal to M_{k}(K).

We end this section showing that every extension inE_{k}has no more than
p conjugates (overk). Of course, the converse is not true, i.e. there exists
an extension of degreep^{2}overkwhich haspconjugates but does not belong
toE_{k}.

Proposition 2.3. Let Lbe an extension of degreep^{2} overk. If there exists
a cyclic extensionK of degreepoverksuch thatK ⊆LandL|Kis Galois,
thenL has no more than p-conjugates (over k).

Proof. Suppose that there exists a cyclic extension K of degree p over k such that K ⊆ L and L|K is Galois. Let L be the normal closure of L

overk: then Gal(L|L) ⊆Gal(L|K) andGal(L|L) is normal in Gal(L|K).

Moreover Gal(L|K) has index p in Gal(L|k). We know that the number of conjugates of L is equal to the index of the normalizer of Gal(L|L) in Gal(L|k). Since the normalizer of Gal(L|L) must contain Gal(L|K), we

see that Lhas 1 or p conjugates.

3. Stucture of Gk(K)

Proposition 3.1. A presentation for G_{k}(K) on the set of generators
{X_{1}, X_{pd+2}} ∪ {X_{l, i}|l= 1, . . . , d, i= 2, . . . , p+ 1}

is given by the relations

X_{l, p+1}^{p} =X_{l, p}^{p} =. . .=X_{l,}^{p}_{2} = 1 l= 1, . . . , d,
X_{pd+2}^{p} =X_{1}, X_{1}^{p} = 1,

[X1, X_{l, i}] = [X_{l, i}, X_{l, j}] = 1

(i, j = 2, . . . , p+ 1,
l= 1, . . . , d,
[X_{l,}_{2}, X_{pd+2}] = [X1, X_{pd+2}] = 1 l= 1, . . . , d,
[Xl, h, Xpd+2] =Xl, h−1

(h= 3,4, . . . , p+ 1, l= 1, . . . , d.

Proof. G_{k}(K) is a group extension of (Z/pZ)^{pd+1} by Z/pZ. We look for
such an extension: let

(3.1) n

X_{l, i}

i= 2, . . . , p+ 1; l= 1, . . . , do

∪ {X_{1}}

be a basis for (Z/pZ)^{pd+1} overFp. The relations above define an automor-
phism σ of (Z/pZ)^{pd+1} (the conjugation by Xpd+2). We have σ^{p}(X) =X
for everyX ∈(Z/pZ)^{pd+1} andσ(X_{1}) =X_{1}. Under these hypotheses, there
exists one and only one extension G of (Z/pZ)^{pd+1} by Z/pZ such that,
for every S ∈ G which represents a generator for the quotient, we have
S^{−1}XS=σ(X) for everyX ∈(Z/pZ)^{pd+1} andS^{p} =X_{1} (see [5]). In other
words inGthe following relations hold:

X_{l, p+1}^{p} =X_{l, p}^{p} =. . .=X_{l,}^{p}_{2}= 1 l= 1, . . . , d,
S^{p} =X_{1}, X_{1}^{p}= 1

[X1, Xl, i] = [Xl, i, Xl, j] = 1 i, j= 2, . . . , p+ 1, l= 1, . . . , d,
[X_{l,}_{2}, S] = [X_{1}, S] = 1 l= 1, . . . , d,

[Xl, h, S] =Xl, h−1 h= 3,4, . . . , p+ 1, l= 1, . . . , d,

whereSis any of the elements which represent a generator for the quotient.

Then the relations of the proposition really define a group G which is an
extension (Z/pZ)^{pd+1} by Z/pZ; moreover G hasd+ 1 generators (look at

the Frattini subgroup). Then (see [3]) there exists a Galois extension E
over k with group G and a Galois extension K of degree p over k such
that E|K is Galois and Gal(E|K) ∼= (Z/pZ)^{pd+1}. Then E =M_{k}(K) and

G=G_{k}(K).

LetH_{k}(K) =hX_{1}, X_{l, i}|i= 2, . . . , p+ 1, l= 1, . . . , di.

Lemma 3.1.

• Every element of order p in Gk(K) belongs to Hk(K): in particular,
G_{k}(K) cannot be written as a semidirect product betweenH_{k}(K)and
a subgroup of order p of G_{k}(K);

• G_{k}(K)^{p}=hX_{1}, X_{l,}_{2}|l= 1, . . . , di;

• [G_{k}(K), G_{k}(K)] =hX_{l, i}|i= 2, . . . , p, l= 1, . . . , di.

Proof. In what follows we consider Hk(K) both as a group and as a vector
space overFp with basis as in (3.1). Let A denote the linear isomorphism
ofH_{k}(K) which corresponds to the conjugation by X_{p+2} on H_{k}(K). If we
defineB =A−I, we have

A^{h}−I =

h

X

i=1

h i

B^{i}
and then

p−1

X

h=0

A^{h}=pI+

p−1

X

h=1 h

X

i=1

h i

B^{i} =

p−1

X

i=1

(

p−1

X

h=i

h i

)B^{i} =B^{p−1}

(for the last equality argue by induction onh fromp−2 to 1 using the well known properties of the binomial coefficient). Let φ the conjugation by Xp+2onHk(K) (so thatAis a description ofφ) and, for everyl= 1, . . . , d,

Xl(n) =Xl(nl,2, nl,3 . . . , nl, p+1) =

p−1

X

i=2

nl, iXl, i. Using the above computations, it is not difficult to see that, if

X=X_{1}^{n}^{1}+

d

X

l=1

X_{l}(n),
then

(X_{p+2}^{n}^{p+2}X)^{p} =X_{p+2}^{n}^{p+2}^{p}

p−1

X

h=0

φ^{n}^{p+2}^{h}X

!

=n_{p+2}X_{1}+

p−1

X

h=0 d

X

l=1

φ^{n}^{p+2}^{h}X_{l}(n)
and, ifp-np+2, the last term is equal to

=n_{p+2}X_{1}+

d

X

l=1

(φ−Id)^{p−1}X_{l}(n)

=n_{p+2}X_{1}+

d

X

l=1

n_{l, p+1}X_{l,}_{2}.

This concludes the proof of the first two claims. The last one follows from
the structure ofG_{k}(K): in fact, if we takeY_{l}∈ hX_{l,}_{2}, X_{l,}_{3}, . . . , X_{l, p+1}i we
haveX_{p+2}^{−j} YlX_{p+2}^{j} ∈ hX_{l,}_{2}, Xl,3, . . . , Xl, pi. SinceX1 belongs to the center

ofGk(K), the last claim follows.

In the case k = Qp, since d = 1, we omit the reference to l in the
generators of G_{p}(K): then we have

Gp(K) =hX_{1}, X2, . . . , Xp+2i.

We putH_{Q}_{p}(K) =H_{p}(K) =hX_{1}, X_{2}, . . . , X_{p+1}i.

Remark. It follows from the Lemma 3.1 thatH_{k}(K) is the only maximal
subgroup G_{k}(K) which is p-elementary abelian. Note that H_{k}(K) is iso-
morphic as anFp[Z/pZ]-module (the action being the conjugation) to the
direct sum of a freeFp[Z/pZ]-module of rank dwith the trivial Fp[Z/pZ]-
module.

4. Normal subgroups and quotients of G_{p}(K).

In what follows we shall focus on the cased= 1, i.e. k=Qp. We denote
by N_{n} the number of normal subgroups of index p^{n} in G_{p}(K) which are
contained inHp(K).

Proposition 4.1. Let A be as in the proof of Lemma 3.1. Then N_{n} (0 <

n ≤ p+ 1) is the number of vector subspaces of dimension p+ 2−n in
Hp(K) invariant under the action of A. Moreover, if 1< n≤p+ 1, then
N_{n}=p+ 1.

Proof. In what follows we consider H_{p}(K) both as subgroup and as vector
space over Fp with basis{X_{1}, X2, . . . , Xp+1}. It is clear that the normal
subgroups of G_{p}(K) with index p^{n} (0 < n ≤ p+ 1) which are contained
inH are exactly the subspaces of Hp(K) of dimensionp+ 2−ninvariant
underA. We claim that the proper subspaces ofH_{p}(K) invariant under A
are exactly those of the form

W_{i}^{λ, µ}={(x_{1}, x_{2}, . . . , x_{p+1})∈(Z/pZ)^{p+1}|λx_{1}+µx_{i}= 0, x_{j} = 0 ifj > i}

where (λ, µ) ∈ Z/pZ×Z/pZ r(0,0), 1 < i ≤ p+ 1 and we used the identification

(x_{1}, x_{2}, . . . , x_{p+1}) =

p+1

X

i=1

x_{i}X_{i}.

First of all, observe that dimW_{i}^{λ, µ} = i−1 and that, if 1 < i ≤ p+ 1 is
fixed, there are exactly p+ 1 distinct W_{i}^{λ, µ}. Then the statement of the
proposition follows from the claim.

Let us prove the claim. On one hand, it is clear that W_{i}^{λ, µ} is in-
variant under A. Conversely, let V be a subspace of H_{p}(K) invariant

under and let k be the maximum of the integers h such that there ex-
ists v = (v_{1}, v_{2}, . . . , v_{p+1}) ∈ V with v_{h} 6= 0 and v_{m} = 0 for m > h:

in particular this means that dimV ≤ k and, if dimV = k ≤ p, then
V =< X_{1}, X_{2}, . . . , X_{k}>=W_{k+1}^{0,}^{1}.

Suppose thatk >2. LetB =A−I: obviouslyV is invariant underB. In
particular (0, v_{k},0, . . . , 0) =B^{k−2}(v)∈V wherev = (v_{1}, v_{2}, . . . , v_{p+1})∈
V is such that v_{k}6= 0. On the other hand

B^{k−3}(v)−vk−1

v_{k} B^{k−2}(v) = (0,0, v_{k},0, . . . , 0)∈V.

Inductively, this means that V contains {X_{i}}^{k−1}_{i=2}. If dimV =k−1, then
we can complete{X_{i}}^{k−1}_{i=2} to form a basis for V adjoining the vector

(v_{1},0, . . . , 0, v_{k},0, . . . , 0).

In this case thenV =W_{k}^{λ, µ}withλ= 1 andµ=−v_{1}/v_{k}. If dimV =k≤p,
we saw thatV =W_{k+1}^{0,1}, while, if dimV =p+ 1, we have V =H_{p}(K).

Suppose now that k = 2: if dimV = 1, then V =< v >= W_{2}^{λ, µ} with
λ= 1 and µ=−v_{1}/v_{2}, otherwise, if the dimension is 2, V =W_{3}^{0,}^{1}.

Lastly, ifk= 1, v= (v_{1},0, . . . ,0) and V =< v >=W_{2}^{0,}^{1}.
In the following we shall denote withW_{j}^{λ, µ}the subgroups defined in the
proof of Prop. 4.1, for 2≤j ≤p+ 1 and (λ, µ)∈ Z/pZ×Z/pZ r(0,0).

Observe thatW_{j}^{λ}^{1}^{, µ}^{1} =W_{j}^{λ}^{2}^{, µ}^{2} if and only if there exists c∈(Z/pZ)^{∗} such
that (λ_{1}, µ_{1}) =c(λ_{2}, µ_{2}).

Lemma 4.1. Let p≥r≥3 and p≥j≥2. Then

G_{p}(K)/W_{j}^{1,}^{0} G_{p}(K)/W_{j}^{λ,}^{1} λ∈Z/pZ,
Gp(K)/W_{r}^{0,}^{1} Gp(K)/W_{r}^{λ,}^{1} λ∈(Z/pZ)^{∗},
Gp(K)/W_{j}^{λ}^{1}^{,}^{1} ∼=Gp(K)/W_{j}^{λ}^{2}^{,}^{1} λ1, λ2 ∈(Z/pZ)^{∗},

G_{p}(K)/W_{2}^{0,}^{1} ∼=G_{p}(K)/W_{2}^{1, µ} µ∈(Z/pZ)^{∗}.

Proof. In order to prove thatG_{p}(K)/W_{j}^{1,}^{0} G_{p}(K)/W_{j}^{λ,}^{1}(λ∈Z/pZ) it is
sufficient to look at the cardinalities of the commutator subgroups of these
two groups. We note that in particularW_{j}^{1,}^{0} is an invariant subgroups of
G_{p}(K).

For the second claim, observe thatG_{p}(K)/W_{r}^{0,}^{1} is regular (it has order
p^{p−r+3} ≤p^{p}, see [1]): in particular it has exponentp (because it admits a
presentation with generators of orderp) while G_{p}(K)/Wr^{λ,}^{1} has exponent
p^{2}. This proves thatGp(K)/Wr^{0,}^{1} Gp(K)/Wr^{λ,}^{1} (λ∈(Z/pZ)^{∗}).

Now, for every λ_{1}, λ_{2} ∈ (Z/pZ)^{∗}, we construct an automorphism σ of
Gp(K) such thatσ(W_{j}^{λ}^{1}^{,}^{1}) =W_{j}^{λ}^{2}^{,}^{1}. Chooseλsuch that< λ >= (Z/pZ)^{∗}

and put

σ(X_{p+2}) =X_{p+2}, σ(X_{1}) =X_{1}, σ(X_{k}) =λX_{k} for 2≤k≤p+ 1.

It is not difficult to verify that this choice defines an automorphism of Gp(K). Moreover for every 2≤j≤p

σ(W_{j}^{1,}^{0}) = (W_{j}^{1,0})
becauseW_{j}^{1,}^{0} is invariant. Then, ifp+ 1≥i≥3,

σ(W_{i}^{λ}^{h}^{,1}) =σ

W_{i−1}^{1,}^{0}⊕< X_{1}−λ^{h}X_{i} >

=W_{i−1}^{1,0}⊕< X1−λ^{h+1}Xi>

=W_{i}^{λ}^{h+1}^{,1}

and we see that our choice of λ gives the result. This proves the third assertion.

Finally, for everyµ∈(Z/pZ)^{∗}, there exists an automorphismτ ofG_{p}(K)
such thatτ(W_{2}^{0,1}) =W_{2}^{1, µ}. In fact, it is easily seen that

(4.1) X_{p+2} 7→X_{p+2}X_{p+1}^{µ} , X_{1} 7→X_{1}X_{2}^{µ}, X_{i} 7→X_{i} ifi >1
effectively defines an automorphism of G_{p}(K) that satisfies the required

properties. This proves the last assertion.

5. Extensions of fields.

Observe that the cardinality of E_{p}(K) is equal to ^{(p}^{p+1}_{p−1}^{−1)}. In fact it
suffices to compute the number of the maximal subgroups ofHp(K) whose
number is precisely ^{(p}^{p+1}_{p−1}^{−1)}.

Using class field theory, it is easily seen that the number of cyclic exten-
sions of degreep^{2} overQp which contain K is equal to p. Moreover, there
is only one Galois extension of degree p^{2} over Qp whose Galois group is
p-elementary abelian. So the normal extensions contained in E_{p}(K) are
exactlyp+ 1.

Now we want to compute the number of extensions in E_{p}(K) whose
normal closure has a fixed group as Galois group. A group which appears
as a Galois group of the normal closure of an extension in E_{p}(K) is a
quotient ofGp(K). The preceding discussion answers the question for the
two groups of orderp^{2} (of course, both of them are quotients ofGp(K)). So
we restrict ourselves to the quotients of orderp^{n}with 3≤n≤p+ 2. In the
following, we denote byE_{j}^{λ, µ}the subextension ofM_{k}(K) which correspond
toW_{j}^{λ, µ}(in the notation for these extensions, we omit the reference toK).

Proposition 5.1. Let 2 ≤ j ≤ p. Then E_{j}^{1,}^{0} is not the splitting field of
any of the extensions in E_{p}(K).

Proof. The result will follow if we prove that every subgroup of orderp^{p−j+1}
contained in the image ofH_{p}(K) inG_{p}(K)/W_{j}^{1,}^{0}contains an element of the
center ofG/W_{j}^{1,0}. In fact, a subgroup of these corresponds to an extension
inE_{p}(K) (E_{j}^{1,}^{0} has degree p^{p−j+3} overQp) and the condition implies that
this extension is contained in a Galois extension of degree strictly less than
p^{p−j+3}. These subgroups correspond to the subgroups of Hp(K) which
contain W_{j}^{1,0} and whose order is p^{p}. Let H^{0} be such a subgroup: suppose
that < X1, Xj+1 >∩H^{0} = {1}. Then H^{0} cannot have order p^{p}. Now we
conclude observing that the image of < X_{1}, X_{j+1} > is contained in the

center ofG/W_{j}^{1,}^{0}.

Proposition 5.2. Let 2≤j≤pand λ∈Z/pZ. Then E_{j}^{λ,}^{1} is the splitting
field of exactly p^{p−j+1} extensions in E_{p}(K).

Proof. First of all, observe that the center ofG_{p}(K)/W_{j}^{λ,}^{1} is generated by
the image of X_{j}. In fact, look at the centralizer of the image of X_{p+2}:
it is easy to see that it is generated by Xj (the bar denotes the images
under the projection) and then it must be equal to < Xj >(the center of
ap-group cannot be trivial). Since in a p-group the intersection between a
normal subgroup and the center is not trivial, we deduce that the center of
G_{p}(K)/W_{j}^{λ,}^{1} is contained in each of his normal subgroups. Now we look
at the subgroups ofH_{p}(K) whose order is p^{p}, which containW_{j}^{λ,}^{1} and do
not contain Xj. These subgroups are in one-to-one correspondence with
the hyperplanes of < X1, Xj, Xj+1 . . . , Xp+1 > which contain the one
dimensional subspace{y_{1}X_{1}+y_{j}X_{j}|λy_{1}+y_{j} = 0}and do not contain X_{j}.
These hyperplanes are exactly the hyperplanes defined by the equations

λy1+yj +cj+1yj+1+. . .+cp+1yp+1= 0.

with cj+1, . . . , cp+1 ∈ Fp. Then the subgroups of Gp(K)/W_{j}^{λ,}^{1} of order
p^{p−j+1} which do not containXj are of the form

(5.1)

Hcj+1, ..., cp+1 ={X_{j}^{z}^{j}· · ·Xp+1
zp+1

|zj+ci+1zj+1+. . .+cp+1zp+1 = 0}.

Observe that these subgroups cannot contain any normal subgroup other-
wise they would contain the center. So it suffices to count the (p−j+ 1)-
tuples of elements ofZ/pZto count all the extensions of degreep^{2} overQp

whose splitting field is E_{j}^{λ,}^{1}.

We may reinterpret these results in the following way. We have p+ 1 Galois extensions andPp

i=2p(p^{p−i+1}) =Pp

j=2p^{j} non-normal extensions in
E_{p}(K). The sum of these two numbers really gives the number of elements

ofE_{p}(K), that is

1 +p+

p

X

j=2

p^{j} =

p

X

j=0

p^{j} = (p^{p+1}−1)
p−1 .

6. Ramification groups of M_{p}(K) when K is unramified.

In the next two sections we are going to use results from local class field theory and ramification theory. We prefer not to report every time the reference: the definitions and the proof of every result concerning those theories can be found in [4].

LetK_{0} be the unramified extension of degreepoverQp: for this section,
as a matter of notation, we putMp(K0) = M0. Let F0 be the unramified
extension of degreep^{2}ofQp: we haveF0 ⊂M0. MoreoverM0|F_{0}is a totally
ramified abelianp-extension (of degreep^{p}). Letψ_{F}^{M}^{0}

0 denotes the Hasse-Arf
function relative to the extension M_{0}|F_{0} and let ϕ^{M}_{F}^{0}

0 be its inverse. We
denote by {G_{i}}and {G^{i}}respectively the lower numbering and the upper
numbering filtrations relative to the extension M0|F_{0}.

Remark. We have

Gal(M_{0}|F_{0}) =hX_{1}^{a}X_{p+1}, X_{2}, X_{3}, . . . , X_{p}i

for somea∈Z/pZsince the subgrouphX_{1}, X2, . . . , Xpihas non cyclic quo-
tient. Therefore up to an automoprhism ofG(more precisely the automor-
phism which fixes every generator except X_{p+1} which maps to X_{1}^{−a}X_{p+1})
we can suppose

Gal(M_{0}|F_{0}) =hX_{2}, X_{3}, . . . , X_{p}, X_{p+1}i.

Proposition 6.1. The following holds: Gal(M_{0}|F_{0}) =G_{0} =G_{1} and G_{i} =
{1} for every i >1.

Proof. We denote as usual for ap-adic fieldF by U_{F}^{i} the subgroup of units
ofF which are congruent to 1 modulo thei-th power of the prime ideal of
F.

First of all, we observe that U_{F}^{2}

0 = (U_{F}^{1}

0)^{p} since F_{0} is (absolutely) un-
ramified. Now, in the isomorphism

F_{0}^{∗}∼=hπ_{F}_{0}i ×F_{p}^{∗}^{2} ×U_{F}^{1}_{0},
the subgroupN_{F}^{M}^{0}

0 (M_{0}^{∗}) corresponds to

N_{F}^{M}_{0}^{0}(M_{0}^{∗})∼=hπ_{F}_{0}i ×F_{p}^{∗}^{2}×N_{F}^{M}_{0}^{0}(U_{M}^{1}_{0}).

Since we haveF_{0}^{∗p} ⊆N_{F}^{M}^{0}

0 (M_{0}^{∗}) (both are normic subgroups and the abelian
extension corresponding to F_{0}^{∗p} containsM0) and we get U_{F}^{2}

0 = (U_{F}^{1}

0)^{p} ⊂
N_{F}^{M}^{0}

0 (U_{M}^{1}

0).

One has G0 =G1 as M0|F_{0} is widly ramified. It follows ϕ^{M}_{F}_{0}^{0}(1) = 1 =
ψ_{F}^{M}^{0}

0 (1). Then from class field theory, we get
U_{F}^{1}_{0}/U_{F}^{2}_{0}N_{F}^{M}^{0}

0 (U_{M}^{1}_{0})∼=G_{1}/G_{2}.
Now, the first term is equal to U_{F}^{1}

0/N_{F}^{M}^{0}

0 (U_{M}^{1}

0) which is isomorphic to

(Z/pZ)^{p}. ThenG_{2} ={1}.

Next we consider the extension M_{0}|Qp. We denote by {G_{i}} and {G^{i}}
respectively the lower numbering and the upper numbering filtrations rela-
tive to the extensionM_{0}|Qp. This notation is consistent with the preceding
one if we restrict ourselves toi≥0, as we will do in the following.

LetW be a normal subgroup of G_{p}(K_{0}) which is contained inH_{p}(K_{0}).

We put

G_{i} = (G_{p}(K_{0})/W)_{i}, G^{i} = (G_{p}(K_{0})/W)^{i}.

We are going to determine the ramification groups of the extension corre-
sponding toW using the well known formulas for the ramification groups
of a quotient. As we are interested in W as long asG_{p}(K_{0})/W is the Ga-
lois group the normal closure of an extension in E_{p}(K0), we can suppose
W 6=W_{j}^{1,}^{0}.

We know thatW is not contained inGal(M0|F_{0}) becauseW is one of the
W_{j}^{λ,}^{1}’s with 2≤j ≤p+ 1 andλ∈Z/pZ. In particular W ·Gal(M0|F_{0}) =
H_{p}(K_{0}). ThenG_{i} =G^{i} =H_{p}(K_{0})/W if 0≤i≤1 and G_{i} =G^{i} ={1} for
everyi >1.

LetE be the extension corresponding toW: we havep^{n}= [E :Qp] form
somen. We denote byd_{E} the discriminant of the extension E|Qp and put
dE =v(dE), where v is the valuation on Qp such that v(p) = 1. Observe
thatW *Gal(M_{0}|F_{0}) impliesF_{0} *E. Then we have

d_{E} = 2p(p^{n−1}−1)

(for this computation we use the Hilbert formula for the different which involves the cardinalities of the ramification groups). Note that the factor pcomes from the inertia index of E|Qp.

7. Ramification groups of Mp(K) when K is totally ramified.

LetK be a totally ramified cyclic extension of degreep overQp and let F be the unramified extension of degree pof K: F is the maximal abelian extension of exponentpoverQp. For this section, as a matter of notation, we put

M_{p}(K) =M, G_{p}(K) =G, H_{p}(K) =H,

every statement of this section being independent of the particular choice of K within the set of cyclic totally ramified extensions of degree p of Qp.

Observe thatF ⊂M andL|M is a totally ramified abelianp-extension (of
degreep^{p}). Letψ_{K}^{M} denotes the Hasse-Arf function relative to the extension
M|K and letϕ^{M}_{K} be its inverse. We denote by{G_{u}}and{G^{v}}respectively
the lower numbering and the upper numbering filtrations relative to the
extension M|Qp. Similarly, we denote by {H_{u}} and {H^{v}} respectively
the lower numbering and the upper numbering filtrations relative to the
extensionM|K. Put finally, for every u and v,

hu=|H_{u}|, h^{v}=|H^{v}|, gu=|G_{u}|, g^{v}=|G^{v}|.

Proposition 7.1. The following holds:

H^{v} ∼= (Z/pZ)^{p−i+1} if i−1< v ≤i, 1≤i≤p−1,
H^{v} ∼=Z/pZ if p−1< v ≤p+ 1

and H^{v} ={1} if v > p+ 1.

Proof. We apply the results of [2] to the extensionM|K: M is the maximal
abelian extension of exponentpoverK andK is totally ramified of degree
poverQp. Then we know that the jumps ofψ^{M}_{K} are 1,2, . . . , p−1, p+ 1,
since

p+ 1< p^{2}

p−1 < p+ 2.

We have f(M|K) =p and e(M|K) =p^{p}, thenh^{0} =h^{1} = p^{p}. Since there
arep jumps, the ratio between the right and the left derivatives of ψ_{K}^{M} at
every jump must be equal top. This proves what we want.

Now observe that for everyi≥ −1 one has

(7.1) H_{i}=G_{i}∩H.

Furthermore

(7.2) ϕM|Qp =ϕK|Qp◦ϕM|K.

Using (7.2) it is easy to computeϕM|Qp and thegi’s. Then we get
g_{0}=g_{1} =p^{p+1},

g_{2+p+...+p}^{j−1} =. . .=g_{1+p+...+p}^{j} =p^{p−j} if 1≤j≤p−2,
g_{2+p+...+p}^{p−2} =. . .=g_{1+p+...+p}^{p−2}_{+2p}^{p−1} =p.

Using (7.1), we can deduce that Hi = Gi, if i ≥ 2: in particular the
subgroups H_{i} are normal in G. Observe that the jumps in the filtration
{G^{v}} arenot integers: more precisely one has

g^{v} =p^{p+1} if 0≤v≤1,
g^{v} =p^{p−i} if 1 +i−1

p < v ≤1 + i

p, i= 1, . . . , p−2,

g^{v} =p if 1 +p−2

p < v≤2,
g^{v} = 1 if v >2.

In the following we shall callvm them-th jump in the filtration{G^{v}}. For
examplev_{1}= 1, v_{2}= 1 + ^{1}_{p} and v_{p} = 2.

Remark. Up to an automorphism ofG(more precisely an automorphism
such thatX_{p+2}X_{p+1}^{h} 7→X_{p+2}, see (4.1)), we can suppose

Gal(L|K_{0}) =hX_{2}, X_{3}, . . . , X_{p}, X_{p+2}i

where we still denote byK0 the unramified extension of degreepof Qp, as in the preceding section.

Lemma 7.1. M|K_{0} has preciselyp cyclic subextensions of degree p^{2}: they
corresponds to the subgroups

hX_{1}^{h}X_{p}, X_{2}, X_{3}, . . . , Xp−1i ⊆Gal(L|K_{0})

ashruns in{0,1, . . . , p−1}. Moreover, ifE one of these subextensions of
M|K_{0} and {Gal(E|K_{0})^{v}} is the upper numbering filtration on Gal(E|K_{0}),
one has

Gal(E|K_{0})^{v} ∼=Z/p^{2}Z if 0≤v≤1
Gal(E|K_{0})^{v} ∼=Z/pZ if 1< v≤2
and Gal(E|K_{0})^{v}={1} if v >2.

Proof. Clear.

Ifh= 0,1, . . . , p−1, we denote with E_{h} the cyclic extension of degree
p^{2} overK0 corresponding tohX_{1}^{h}Xp, X2, X3, . . . , Xp−1i. Observe that
(7.3) (G/Gal(M|E_{h}))^{v} =G^{v}Gal(M|E_{h})/Gal(M|E_{h}).

Proposition 7.2. There exists m∈Z/pZ such that the following holds
G^{v} =hX_{2}, X3, . . . , Xp, Xp+2i if 0≤v≤v1

G^{v} =hX_{1}, X_{2}, . . . , Xp−ii if

(v_{i}< v ≤v_{i+1},
i= 1, . . . , p−2
G^{v} =hX_{1}X_{2}^{m}i if vp−1 < v≤v_{p}
and G^{v} ={1} if v > v_{p}.

Proof. The first claim is clear because

G^{1} =G^{0}=Gal(M|K_{0}).

Now observe that Gal(M|K_{0})^{v} = G^{v} if v ≥ 0. Using (7.3) we get, if
1< v ≤2, for every 0≤h≤p−1,

Z/pZ∼= (Gal(M|K_{0})/Gal(E_{h}|K_{0}))^{v}

=G^{v}hX_{1}^{h}X_{p}, X_{2}, . . . , Xp−1i/hX_{1}^{h}X_{p}, X_{2}, . . . , Xp−1i.

(7.4)

Now, G^{v} is p-elementary abelian (since we showed that G_{i} =H_{i} ifi≥ 2)
and it is contained in G^{1}. Then, sinceg^{v} =p^{p−1} if 1< v≤v2,

G^{v} =hX_{1}, X_{2}, . . . , Xp−1i if 1< v≤v_{2}

(one can also use the fact that G^{1}/G^{v}^{1} has to be a p-elementary abelian
group). Observe that g^{v}^{3} = p^{p−2}; furthermore G^{v}^{3} 6= hX_{2}, . . . , Xp−1i
because of (7.4). Then G^{v}^{3} =hX_{1}X_{p−1}^{l} X2, . . . , Xp−2i for some l∈Z/pZ.

We have

(7.5) hX_{p−2}^{l} , X_{2}, . . . , Xp−3i= [G^{1}, G^{v}^{3}]⊆G^{v}^{4}

If l 6= 0, hX_{p−2}^{l} , X2, . . . , Xp−3i has order p^{p−3} = g^{v}^{4}: then in (7.5), we
would have equality. But this is impossible becauseG^{v}^{4} *hX_{2}, . . . , Xp−1i
since (7.4) holds. ThenG^{v}^{3} =hX_{1}, X_{2}, . . . , Xp−2i.

In a similar way one proves that

G^{v} =hX_{1}, X2, . . . , Xp−ii

ifvi < v≤vi+1andi= 1, . . . , p−2. ThenG^{v}^{p−1} =hX_{1}, X2ibut we cannot
use the commutator argument again because bothX_{1}andX_{2}belong to the
center ofG. Still, thanks to (7.4), we haveG^{v}^{p} 6=hX_{2}i. This concludes the

proof.

For everyW_{j}^{λ, µ} we put

(G^{λ, µ}_{j} )^{v} =

G/W_{j}^{λ, µ}
v

.

As in the preceding section we are going to determine the ramification
groups of the extension E_{j}^{λ, µ} overQp corresponding to W_{j}^{λ, µ}. We denote
by d_{E}λ, µ

j |Qp the discriminant of the extension E_{j}^{λ, µ}|Qp and put d_{E}^{λ, µ}

j |Qp =
v(d_{E}λ, µ

j

), where v is the valuation on Qp such that v(p) = 1. We suppose
2 ≤ j ≤ p, because {E_{p+1}^{λ, µ}|Qp} is the set of Galois extensions of degree
p^{2} over Qp, whose discriminants are well known. Furthermore, as we are
interested in theE_{j}^{λ, µ} as long as they are the normal closure of extensions
of degree p^{2} over Qp, we are going to omit the computations for the case
µ= 0 and λ= 1 (see Prop. 5.1).

Suppose firstµ= 1,λ= 0 and 3≤j ≤p. Then we have

|(G^{λ, µ}_{j} )^{v}|=p^{p−j+2} if 0≤v≤v1,

|(G^{λ, µ}_{j} )^{v}|=p^{p−j−i+1} if vi < v≤vi+1, 1≤i≤p−j,
and |(G^{λ, µ}_{j} )^{v}|= 1 if v > vp−j+1. Then

d_{E}^{0,}^{1}

j |Qp =p 2(p^{p−j+2}−1) +

p−j

X

i=1

p^{i}(p^{p−j−i+1}−1)

! . Now suppose µ= 1,λ6= 0 and 3≤j≤p−1. Then we have

|(G^{λ, µ}_{j} )^{v}|=p^{p−j+2} if 0≤v≤v1,

|(G^{λ, µ}_{j} )^{v}|=p^{p−j−i+1} if v_{i} < v≤v_{i+1}, 1≤i≤p−j−1,

|(G^{λ, µ}_{j} )^{v}|=p if vp−j < v ≤vp,
and |(G^{λ, µ}_{j} )^{v}|= 1 if v > vp. Then

d_{E}λ,1

j |Qp =p

2(p^{p−j+2}−1) +

p−j−1

X

i=1

p^{i}(p^{p−j−i+1}−1)+

+p^{p−j}(vp−vp−j)(p−1)

. Ifµ= 1,λ6= 0 and j=p, then we have

|(G^{λ, µ}_{j} )^{v}|=p^{2} if 0≤v≤v_{1},

|(G^{λ, µ}_{j} )^{v}|=p if v_{1} < v≤v_{p}
and |(G^{λ, µ}_{j} )^{v}|= 1 if v > vp. Then

d_{E}λ,1

p |Qp =p 2(p^{2}−1) +p(p−1)
.

Now suppose j = 2. Observe that we have G/W_{2}^{0,}^{1} ∼= G/W_{2}^{1, µ} for every
µ∈(Z/pZ)^{∗}(see Lemma 4.1). We have two cases which must be considered
separately. Suppose first thatW_{2}^{λ, µ}=G^{v}^{p} =hX_{1}X_{2}^{m}i: then, denoting with
de the valuation of the discriminant of E_{2}^{λ, µ}|Qp in this case, we have

de=p 2(p^{p}−1) +

p−2

X

i=1

p^{i}(p^{p−i−1}−1)

! .

Conversely suppose thatW_{2}^{λ, µ}6=G^{v}^{p}: then, denoting withduthe valuation
of the discriminant ofE^{λ, µ}_{2} |Qp in this case, we have

du =p 2(p^{p}−1) +

p−3

X

i=1

p^{i}(p^{p−i−1}−1) + (vp−vp−2)p^{p−1}(p−1)

! .

Now we look at the extensions E_{j}^{λ, µ}|L where L ∈ E_{p}(K), µ 6= 0 and
2 ≤ j ≤ p: we have described the Gal(E_{j}^{λ, µ}|L)’s in Prop. 5.2 (see 5.1).

They are subgroups ofG^{λ, µ}_{j} and then, for every i,

Gal(E_{j}^{λ, µ}|L)

i =
G^{λ, µ}_{j}

i

∩Gal(E_{j}^{λ, µ}|L).

We denote by d

E_{j}^{λ, µ}|L the discriminant of E_{j}^{λ, µ}|L and we put d_{E}λ, µ
j |Qp =
w(d_{E}λ, µ

j |Qp), where w is the valuation on L such that w|v and w(p) =p^{2}.
Suppose firstµ= 1, λ= 0 and 3≤j ≤p. Then we get

d_{E}^{0,}^{1}

j |L=p 2(p^{p−j} −1) +

p−j

X

i=1

p^{i}(p^{p−j−i}−1)

! . Now suppose µ= 1,λ6= 0 and 3≤j≤p. Then we get

d_{E}λ,1

j |L=p 2(p^{p−j}−1) +

p−j

X

i=1

p^{i}(p^{p−j−i}−1)

!
.
Finally, we analyze the case j = 2: it easily seen that d_{E}λ, µ

2 |L does not
depend on whetherW_{2}^{λ, µ}=G^{v}^{p} or not. Furthermore we have

d2|L=d_{E}^{λ, µ}

2 |L=p 2(p^{p−2}−1) +

p−3

X

i=1

p^{i}(p^{p−i−2}−1)

! .

We denote byd_{L|}_{Q}_{p} the discriminant ofL|Qp and we putd_{L|}_{Q}_{p} =v(d_{L|}_{Q}_{p}).

Using the formula for the discriminants in towers of extensions, we get, if 3≤j≤p,

d_{E}λ, µ

j |Qp=p^{p−j+1}d_{L|}_{Q}_{p}+d_{E}λ, µ
j |L

and analogous formulas in the case j= 2.

8. Classification table

We collect our results in a table which describes the classification of the
extensions of degreep^{2} overQp whose normal closure is a p-extension. We
recall some notations. Let λ be a fixed element in (Z/pZ)^{∗} and let j run
in {3,4, . . . , p}. In the following table, the first four lines list the Galois
extensions of degreep^{2} overQp. Lines from the fifth to the seventh list the
non-normal extension of E_{Q}_{p}(K_{0}). The remaining lines describe the non
normal totally ramified extensions of degreep^{2} overQp. When j appears
in a line, it simply means that there is a set of lines obtained by replacingj
with 3,4, . . . , p. For an extensionL∈ E_{Q}_{p},e=e(L|Qp) is the ramification
index of L, d = dL|Qp = v(dL|Qp) is the valuation of the discriminant of
L, G = Gal(L|Qp) where L is the normal closure of L over Qp and we