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: [email protected] (DucVanHuynh),[email protected] (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 extensionsofap-adicfield.J.NumberTheory131(2011),no.8,1429–1434. 9