k-Isomorphism Classes of Local Field Extensions

0.13 MB·English

by  Duc Van Huynh

k-isomorphism classes of local field extensions
Duc Van Huynh1, Kevin Keating
Department of Mathematics, Universityof Florida, Gainesville, FL 32611-8105, USA
5
1
0
2

n
a Abstract
J

2 LetK bealocalfieldofcharacteristicpwithperfectresiduefieldk. Inthispaper
2 wefind asetofrepresentativesforthe k-isomorphismclassesoftotally ramified

separable extensions L/K of degree p. This extends work of Klopsch, who
]
T found representatives for the k-isomorphism classes of totally ramified Galois
N extensions L/K of degree p.
.
h
t
a 1. Introduction and results
m
[ Let K be a local field with perfect residue field k and let Ks be a separable
closure of K. The problem of enumerating finite subextensions L/K of Ks/K
1
has a long history (see for instance [5]). Alternatively, one might wish to enu-
v
merate isomorphism classes of extensions. Say that the finite extensions L /K
2 1
3 and L2/K are K-isomorphic if there is a field isomorphism σ :L1 →L2 which
5 induces the identity map on K. In this case the extensions L /K and L /K
1 2
5 share the same field-theoretic and arithmetic data; for instance their degrees,
0
automorphism groups, and ramification data must be the same. In the case
.
1 where K is a finite extension of the p-adic field Q , Monge [6] computed the
p
0
number of K-isomorphism classes of extensions L/K of degree n, for arbitrary
5
n≥1.
1
: One says that the finite extensions L1/K and L2/K are k-isomorphic if
v
there is a field isomorphism σ : L → L such that σ(K) = K and σ induces
i 1 2
X the identity map on k. Such an isomorphism is automatically continuous (see
r Lemma 3.1). If the extensions L1/K and L2/K are k-isomorphic then they
a have the same field-theoretic and arithmetic properties. Let Aut (K) denote
k
the group of field automorphisms of K which induce the identity map on k.
Then Aut (K) is finite if char(K) = 0, infinite if char(K) = p. Since every
k
k-isomorphism σ from L /K to L /K induces an element of Aut (K), this
1 2 k
suggests that k-isomorphisms should be more plentiful when char(K) = p. In
Email addresses: [e-mail] (DucVanHuynh),[e-mail]
(KevinKeating)
1Current address: Department of Mathematics, Armstrong State University, Savannah,
GA31419
Preprint submitted toElsevier January 23, 2015
thispaperweconsiderthe problemofclassifyingk-isomorphismclassesoffinite
totally ramified extensions of a local field K of characteristic p.
As one might expect, the tame case is straightforward: It is easily seen
that if n ∈ N is relatively prime to p then there is a unique k-isomorphism
class of totally ramified extensions L/K of degree n. We will focus on ramified
extensions of degree p, which are the simplest non-tame extensions. Since any
two k-isomorphic extensions have the same ramification data, it makes sense
to classify k-isomorphism classes of degree-p extensions with fixed ramification
break b>0.
LetE denotethesetofalltotallyramifiedsubextensionsofK /K ofdegree
b s
p with ramification break b, and let S denote the set of k-isomorphism classes
b
of elements of E . Let Sg denote the set of k-isomorphism classes of Galois
b b
extensions in E , and let Sng denote the set of k-isomorphism classes of non-
b b
Galois extensions in E . As we will see in Section 2, if b is the ramification
b
break of an extension of degree p then (p−1)b∈NrpN. Hence S is empty if
b
b6∈ 1 ·(NrpN).
p−1
Theorem1.1. Letb∈ 1 ·(NrpN)andwriteb= (m−1)p+λ with1≤λ≤p−1.
p−1 p−1
LetR={ω :i∈I}beasetofcosetrepresentativesfork×/(k×)(p−1)b. Foreach
i
ω ∈R let π ∈K be a root of the polynomial Xp−ω πmXλ−π . Then the
i i s i K K
mapwhichcarriesω ontothek-isomorphismclassofK(π )/K givesabijection
i i
from R to S . Furthermore, K(π )/K is Galois if and only if b ∈ NrpN and
b i
λω ∈(k×)p−1.
i
Corollary 1.2. Let b∈ 1 ·(NrpN) and assume that |k|=q <∞. Then
p−1
|S |=gcd(q−1,(p−1)b).
b
Furthermore, if b∈NrpN then
q−1
|Sg|=gcd ,b
b p−1
(cid:18) (cid:19)
q−1
|Sng|=(p−2)·gcd ,b .
b p−1
(cid:18) (cid:19)
Proof. This follows from Theorem 1.1 and the formulas
|k×/(k×)(p−1)b|=gcd(q−1,(p−1)b),
q−1
|(k×)p−1/(k×)(p−1)b|=gcd ,b for b∈NrpN.
p−1
(cid:18) (cid:19)
TheproofofTheorem1.1reliesheavilyontheworkofAmano,whoshowedin
[1] that every degree-p extension of a local field of characteristic 0 is generated
by a root of an Eisenstein polynomial with a special form, which we call an
Amano polynomial (see Definition 2.4). In Section 2 we show how Amano’s
results can be adapted to the characteristic-p setting. In Section 3 we prove
Theorem 1.1 by computing the orbits of the action of Aut (K) on the set of
k
Amano polynomials over K.
2
2. Amano polynomials in characteristic p
Let F be a finite extension of the p-adic field Q and let E/F be a totally
p
ramified extension of degree p. In [1], Amano constructs an Eisenstein polyno-
mial g(X) over F with at most 3 terms such that E is generated over F by a
root of g(X). In this section we reproduce a part of Amano’s construction in
characteristicp. Weassociateafamilyof3-termEisensteinpolynomialstoeach
ramifiedseparable extension of L/K of degree p, but we don’t choose represen-
tatives for these families. Many of the proofs from [1] remain valid in this new
setting.
Let K be a local field of characteristic p with perfect residue field k. Let
K be a separable closure of K and let ν be the valuation of K normalized
s K s
so that ν (K×)=Z. Fix a prime element π for K; since k is perfect we may
K K
identify K with k((π )). Let U denote the group of units of K, and let U
K K 1,K
denote the subgroupof 1-units. If u∈U and α∈Z is a p-adic integer then
1,K p
uα isdefinedasalimitofpositiveintegerpowersofu. Thisappliesinparticular
when α is a rational number whose denominator is not divisible by p.
Let L/K be a finite totally ramified subextension of K /K and let ν be
s L
the valuationofK normalizedsothatν (L×)=Z. Letπ be aprimeelement
s L L
for L and let σ : L → K be a K-embedding of L into K , such that σ 6= id .
s s L
We define the ramification number of σ to be ν (σ(π )−π )−1. It is easily
L L L
seenthatthis definitiondoesnotdependonthe choiceofπ . Wesaythatbisa
L
(lower)ramificationbreak of the extension L/K if b is the ramificationnumber
of some nonidentity K-embedding of L into K .
s
Suppose L/K is a separable totally ramified extension of degree p. Then
Lemma 1 of [1] shows that L/K has a unique ramification break. Every prime
element π of L is a root of an Eisenstein polynomial
L
p−1
f(X)=Xp− c Xi
i
i=0
X
over K, with ν (c ) = 1 and ν (c ) ≥ 1 for 1 ≤ i ≤ p−1. Let π′ 6= π be a
K 0 K i L L
conjugate of π in K . Then the ramification break of L/K is given by
L s
′
π
b=ν L −1 .
L
π
(cid:18) L (cid:19)
Since L/K is separable, we have c 6= 0 for some i with 1 ≤ i ≤ p−1.
i
Therefore
m=min{νK(c1),...,νK(cp−1)}
is finite. Let λ be minimum such that ν (c ) = m and let ω ∈ k× satisfy
K λ
c ≡ ωπm (mod πm+1). We say that the Eisenstein polynomial f(X) is of
λ K K
type hλ,m,ωi. Note that while ω depends on the choice of π , the positive
K
integers m and λ do not. If f(X) is of type hλ,m,ωi then by Lemma 1 of [1]
the ramification break b of L/K is given by
(m−1)p+λ
b= . (2.1)
p−1
3
Conversely,givenb∈ 1 ·(NrpN), equation(2.1) uniquely determines m and
p−1
λ,andwecaneasilyconstructEisensteinpolynomialsoftypehλ,m,ωiforevery
ω ∈k×.
For Eisenstein polynomials f(X),g(X)∈K[X], write f(X)∼g(X) if there
is a K-isomorphism
K[X]/(f(X))∼=K[X]/(g(X)).
Then ∼ is an equivalence relation on Eisenstein polynomials over K.
Theorem 2.1. Suppose f(X),g(X)∈K[X] are Eisenstein polynomials of de-
gree p such that f(X)∼g(X). Then f(X) and g(X) are of the same type.
Proof. The proof of Theorem 1 of [1] applies here, except that in characteristic
p we don’t have to consider polynomials of type h0i.
Henceforth we say that an extension L/K has type hλ,m,ωi if L/K is
K-isomorphic to K[X]/(f(X)) for some Eisenstein polynomial f(X) of type
hλ,m,ωi.
Theorem 2.2. LetL/K beanextensionoftypehλ,m,ωi. ThenL/K isGalois
if and only b= (m−1)p+λ is an integer and λω ∈(k×)p−1.
p−1
Proof. The proof of Theorem 3(ii) of [1] applies without change.
Theorem 2.3. Suppose L/K is an extension of type hλ,m,ωi. Then there
exists a prime element π ∈L which is a root of a polynomial
L
Ab (X)=Xp−ωπmXλ−uπ
ω,u K K
for some u∈U .
1,K
Proof. The proofofTheorem4 of[1]applies here,exceptthat we don’thaveto
consider extensions of type h0i. Briefly, one defines a function φ:L→K by
φ(α)=αp−ωπmαn−N (α),
K L/K
where N is the norm from L to K. Using an iterative procedure one gets a
L/K
prime element π in L such that ν (φ(π)) > p(λ+1) and N (π) = uπ for
L L/K K
some u∈U . Let π(1),...,π(p) ∈K be the roots of Ab (X). Then
1,K s ω,u
p
φ(π)=Ab (π)= (π−π(i)), (2.2)
ω,u
i=1
Y
so we have
p
ν (π−π(i))=ν (φ(π))>p(λ+1). (2.3)
L L
i=1
X
Hence ν (π −π(j)) > λ+1 for some j, so we get L ⊂ K(π(j)) by Krasner’s
L
Lemma. Since [K(π(j)) : K] = [L : K] = p, it follows that L = K(π(j)).
Therefore π =π(j) satisfies the conditions of the theorem.
L
4
Definition 2.4. We say that Ab (X) is an Amano polynomial over K with
ω,u
ramification break b.
Let b = (m−1)p+λ with 1 ≤ λ ≤ p − 1. We denote the set of Amano
p−1
polynomials over K with ramification break b by
P ={Xp−ωπmXλ−uπ :ω ∈k×, u∈U }.
b K K 1,K
Let P /∼ denote the set of equivalence classes of P with respect to ∼. For
b b
f(X)∈P , we denote the equivalence class of f(X) by [f(X)]. It follows from
b
Theorem 2.3 that these equivalence classes are in one-to-one correspondence
with the elements of E .
b
3. The action of Autk(K) on extensions
In this section we show how Aut (K) acts on the set of equivalence classes
k
of Amano polynomials with ramification break b. We determine the orbits of
this action,andgivearepresentativeforeachorbit. Thisallowsusto construct
representatives for the elements of S , and leads to the proof of Theorem 1.1.
b
The following lemma is certainly well-known (see, for instance, the answers
to [3]), but we could find no reference for it.
Lemma 3.1. Let L and L be local fields. Assume that L and L have
1 2 1 2
the same residue field k, and that k is a perfect field of characteristic p. Let
σ :L →L be a field isomorphism. Then ν ◦σ =ν .
1 2 L2 L1
Proof. ThegroupU isn-divisibleforallnprimetop,sowehaveσ(U )⊂
1,L1 1,L1
U . For i = 1,2 the group T of nonzero Teichmu¨ller representatives of L is
eqLu2alto ∞ (L×)pi, so we havieσ(T )=T . Since U =T ·U this impliies
i=1 i 1 2 Li i Li,1
σ(U ) ⊂ U . The same reasoning shows that σ−1(U ) ⊂ U , so we get
L1 T L2 L2 L1
σ(U ) = U . It follows that ν ◦σ, like ν , induces an isomorphism of
L×/LU1 ontoL2Z. Let π be a primL2e element ofL1L . Then 1+π ∈ U , so
1 L1 L1 1 L1 L1,1
ν (σ(1+π ))=0. Hence ν (σ(π ))≥0. Since ν (σ(π )) generatesZ, it
L2 L1 L2 L1 L2 L1
follows that ν (σ(π ))=1. We conclude that ν ◦σ =ν .
L2 L1 L2 L1
For f(X) ∈ K[X] and ϕ ∈ Aut (K) we let fϕ(X) denote the polynomial
k
obtained by applying ϕ to the coefficients of f(X). The following lemma is a
straightforward“transport of structure” result:
Lemma 3.2. Let f(X) and g(X) be Eisenstein polynomials with coefficients
in K such that f(X)∼g(X), and let ϕ∈Aut (K). Then fϕ(X)∼gϕ(X).
k
Let A = Aut (K) denote the group of k-automorphisms of K. Since all
k
k-automorphisms of K = k((π )) are continuous by Lemma 3.1, every ϕ ∈ A
K
is determined by the value of ϕ(π ). Furthermore, A acts transitively on the
K
set of prime elements of K. It follows that the group consisting of the power
series
∞
a ti :a ∈k, a 6=0
i i 1
( )
i=1
X
5
withtheoperationofsubstitutionisisomorphictotheoppositegroupAopofA.
Foreveryϕ∈A therearel ∈k× andv ∈U suchthatϕ(π )=l ·v ·π .
ϕ ϕ 1,K K ϕ ϕ K
Let
N ={σ ∈A :σ(π )∈U ·π }
K 1,K K
be the group of wild automorphisms of K. Then N op is isomorphic to the
Nottingham Group over k (see [4]). Furthermore, N is normal in A, and
A/N ∼=k×.
Letϕ∈A andletAb (X)∈P . ThenbyTheorem2.3thereexistω′ ∈k×
ω,u b
′
and u ∈U such that
1,K
K[X]/((Ab )ϕ(X))=K[X]/(Xp−ϕ(ωπm)Xλ−ϕ(π u))
ω,u K K
∼=K[X]/(Abω′,u′(X)).
It follows from Lemma 3.2 that
ϕ·[Abω,u(X)]=[Abω′,u′(X)] (3.1)
gives a well-defined action of A on P /∼. The following theorem computes
b
explicit values for ω′ and u′ in (3.1). Note that since k is perfect, l has a
ϕ
1
unique pth root lp in k.
ϕ
Theorem3.3. Letϕ∈A andAb (X)∈P . Thenϕ·[Ab (X)]=[Ab (X)],
ω,u b ω,u ω′,u′
with ω′ =ω·l(p−p1)b, u′ =ϕ(u)·vh, and h= p−λ−pm.
ϕ ϕ p−λ
Proof. By applying ϕ to the coefficients of Ab (X) we get
ω,u
(Ab )ϕ(X)=Xp−ωlmvmπmXλ−ϕ(u)l v π .
ω,u ϕ ϕ K ϕ ϕ K
1 m
Set X =lpvp−λZ. Then
ϕ ϕ
l−1v−p−pmλ(Ab )ϕ(X)=Zp−ωl(p−p1)bπmZλ−ϕ(u)vhπ
ϕ ϕ ω,u ϕ K ϕ K
=Zp−ω′πmZλ−u′π .
K K
1 m
Since lpvp−λ ∈K, it follows that
ϕ ϕ
K[X]/(Abω,u)ϕ(X))∼=K[X]/(Abω′,u′(X)).
To determine the orbit of [Ab (X)] under the action of A we need the
ω,u
following lemmas. Let Z× denote the unit group of the ring of p-adic integers.
p
Lemma 3.4. Let u∈U , and h∈Z×. Then
1,K p
σ(π ) h
U = σ(u)· K : σ ∈N .
1,K
π
( (cid:18) K (cid:19) )
6
Proof. Let v =uh1 ∈U1,K. Then πK′ =vπK is a prime element of K. We have
′
vσ(π )
U = K :σ ∈N
1,K π′
(cid:26) K (cid:27)
σ(vπ )
= K :σ ∈N
π
(cid:26) K (cid:27)
σ(π )
= σ(u)h1 · K :σ ∈N .
π
(cid:26) K (cid:27)
Sinceh∈Z×,wehaveUh =U . Hence byraisingtothepowerh weobtain
p 1,K 1,K
h
σ(π )
U = σ(u)· K : σ ∈N .
1,K
π
( (cid:18) K (cid:19) )
Lemma 3.5. Let c∈k× and define τ ∈A by τ (π )=cπ . Let N =N τ
c c K K c c
be the rightcosetofN inA representedby τ . Thenforu∈U andh∈Z×
c 1,K p
we have
U ={ϕ(u)·vh :ϕ∈N }.
1,K ϕ c
Proof. Let u′ =τ (u)∈U . Then
c 1,K
{ϕ(u)·vh :ϕ∈N }={στ (u)·vh :σ ∈N }
ϕ c c σ
={σ(u′)·vh :σ ∈N }
σ
=U ,
1,K
where the last equality follows from Lemma 3.4.
Theorem 3.6. The orbit of [Ab (X)] under A is
ω,u
A ·[Ab (X)]={[Ab (X)]:θ ∈(k×)(p−1)b, v ∈U }.
ω,u ωθ,v 1,K
Proof. Let c∈k× and ϕ∈N . Then l =c, so by Theorem 3.3 we have
c ϕ
ϕ·[Abω,u(X)]=[Aω′,u′],
with ω′ = ωc(p−p1)b, u′ = ϕ(u)vϕh, and h = p−pλ−−λpm. Hence by Lemma 3.5 we
have
Nc·[Abω,u(X)]={[Aω′,v]:ω′ =ωc(p−p1)b, v ∈U1,K}.
Since A is the union of N over all c ∈ k×, and k is perfect, the theorem
c
follows.
We now give the proof of Theorem 1.1. Let R = {ω : i ∈ I} be a set of
i
coset representatives for k×/(k×)(p−1)b. For each ω ∈R let π ∈K be a root
i i s
of the Amano polynomial
Ab (X)=Xp−ω πmXλ−π .
ωi,1 i K K
7
It follows from Theorem 3.6 that for every equivalence class C ∈ S there is
b
i ∈ I such that K(π )/K ∈ C. On the other hand, if K(π )/K is k-isomorphic
i i
to K(π )/K then by Theorem 3.3, for some ϕ∈A we have
j
[Abωj,1(X)]=ϕ·[Abωi,1(X)]=[Abωi′,vϕh(X)].
(p−1)b
with ω′ =ω l p . It follows from Theorem 2.1 that Ab (X) and Ab (X)
i i ϕ ωj,1 ωi′,vϕh
have the same type, so we have ω =ω l(p−p1)b. Since lp1 ∈k×, this implies that
j i ϕ ϕ
ω ω−1 ∈(k×)(p−1)b. Sinceω andω arecosetrepresentativesfork×/(k×)(p−1)b,
j i i j
we get ω = ω . This proves the first part of Theorem 1.1. The second part
i j
follows from Theorem 2.2.
Remark3.7. In[4],Klopschusesadifferentmethodtocomputethecardinality
of Sg. Let L = k((π )) be a local function field with residue field k, and
b L
set F = Aut (L). Then there is a one-to-one correspondence between cyclic
k
subgroups G ≤ F of order p and subfields M = LG of L such that L/M is
a cyclic totally ramified extension of degree p. For i = 1,2 let G be a cyclic
i
subgroupofF oforderpandsetKi =LGi. SaytheextensionsL/K1andL/K2
are k∗-isomorphic if there exists η ∈F =Aut (L) such that η(K )=K ; this
k 1 2
is equivalent to η−1G η =G .
1 2
For i = 1,2 let ψ : K → L be a k-linear field embedding such that
i
ψ (K) = K . We can use ψ to identify K with K , which makes L an ex-
i i i i
tension of K. We easily see that the extensions ψ : K ֒→ L and ψ : K ֒→ L
1 2
are k-isomorphic if and only if L/K and L/K are k∗-isomorphic. Therefore
1 2
classifyingk-isomorphismclassesofdegree-pGaloisextensionsofK isequivalent
to classifying conjugacy classes of subgroups of order p in F.
For i=1,2 let G =hγ i. If G and G have ramification break b then
i i 1 2
γ (π )≡π +r πb+1 (mod πb+2)
1 L L b+1 L L
γ (π )≡π +s πb+1 (mod πb+2)
2 L L b+1 L L
for some r ,s ∈k×. Hence for 1≤j ≤p−1, we have
b+1 b+1
γj(π )≡π +jr πb+1 (mod πb+2).
1 L L b+1 L L
By Proposition 3.3 of [4], γj and γ are conjugate in F if and only if s =
1 2 b+1
jr tb, for some t∈k×. Therefore the subgroups G and G are conjugate in
b+1 1 2
F ifandonlyifs ∈r ·F×·(k×)b. Itfollowsthatthenumberofconjugacy
b+1 b+1 p
classes of subgroups of order p with ramification break b is
|k×/(F×·(k×)b)|=|(k×)p−1/(k×)(p−1)b|.
p
q−1
In particular, if |k| = q < ∞ then there are gcd ,b such conjugacy
p−1
(cid:18) (cid:19)
classes, in agreement with Corollary 1.2.
8
References
[1] Shigeru Amano, Eisenstein equations of degree p in a p-adic field, J. Fac.
Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 1–21.
[2] IvanFesenkoandSergeiVostokov,LocalFieldsandTheirExtensions,Trans-
lation of Mathematical Monographs V. 121, (AMS, 2002).
[3] Kevin Keating, Automorphisms of k((X)),
http://mathoverflow.net/questions/193757(2015).
[4] Benjamin Klopsch, Automorphisms of the Nottingham Group, Journal of
Algebra 223 (2000) 37–56.
[5] MarcKrasner,Nombredesextensionsd’undegr´edonn´ed’uncorpsp-adique.
(French) 1966 Les Tendances G´eom. en Alg`ebre et Th´eorie des Nombres
pp.143–169EditionsduCentreNationaldelaRechercheScientifique,Paris.
[6] Maurizio Monge, Determination of the number of isomorphism classes of
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