Constructing class fields over local fields

Journal de Th´eorie des Nombres de Bordeaux 18(2006), 627–652

Constructing class fields over local fields

parSebastian Pauli

Dedicated to Michael Pohst on his 60th Birthday R´esum´e. Soit K un corps p-adique. Nous donnons une carac- t´erisation explicite des extensions ab´eliennes de K de degr´epen reliant les coefficients des polynˆomes engendrant les extensions L/K de degr´e p aux exposants des g´en´erateurs du groupe des normes N_L/K(L^∗). Ceci est appliqu´e `a un algorithme de con- struction des corps de classes de degr´e p<sup>m</sup>, ce qui conduit `a un algorithme de calcul des corps de classes en g´en´eral.

Abstract. LetK be a p-adic field. We give an explicit charac- terization of the abelian extensions ofKof degreepby relating the coefficients of the generating polynomials of extensionsL/Kof de- greepto the exponents of generators of the norm groupN_L/K(L^∗).

This is applied in an algorithm for the construction of class fields of degree p^m, which yields an algorithm for the computation of class fields in general.

1. Introduction

Local class field theory gives a complete description of all abelian ex- tensions of a p-adic field K by establishing a one-to-one correspondence between the abelian extensions of K and the open subgroups of the unit group K^∗ of K. We describe a method that, given a subgroup of K^∗ of finite index, returns the corresponding abelian extension.

There are two classic approaches to the construction of abelian exten- sions: Kummer extensions and Lubin-Tate extensions. Kummer extensions are used in the construction of class fields over global fields [Fie99, Coh99].

The theory of Lubin-Tate extensions explicitly gives generating polynomials of class fields overp-adic fields including the Artin map.

The goal of this paper is to give an algorithm that constructs class fields as towers of extensions from below thus avoiding the computation of a larger class field and the determination of the right subfield. The wildly ramified

Manuscrit re¸cu le 31 d´ecembre 2005.

part of a class field is constructed as a tower of extensions of degreep over the tamely ramified part of the class field.

Our approach allows the construction of class fields of larger degree than the approach with Lubin-Tate or Kummer extensions. Given a subgroupG ofK^∗these methods provide a class fieldLH that corresponds to a subgroup H of G and that contains the class field corresponding to G. In general the degree of LH is very large and the computation of the corresponding subfield expensive. Our approach does not yield a construction of the Artin map though.

We start by recalling the structure of the unit groups of p-adic fields (section 2). In section 3 we state the main results of local class field theory and the explicit description of tamely ramified class fields. It follows that we can restrict our investigation to cyclic class fields of degreep^m. We begin our investigation by constructing a minimal set of generating polynomials of all extensions of K of degree p (section 4). In section 5 we relate the coefficients of the polynomials generating extensions of degree p to the exponents of the generators of their norm groups. This yields an algorithm for computing class fields of degreep. Section 6 contains an algorithm for computing class fields of degree p^m. In section 7 we give several examples of class fields.

Given a fixed prime number p, Qp denotes the completion of Q with respect to the p-adic valuation | · | = p^−νp(·), K is a finite extension of degree n overQp complete with respect to the extension of | · | toK, and O_K = {α∈K | |α|61} is the valuation ring of K with maximal ideal p_K = {α∈K| |α|<1} = (π_K). The residue class field is defined by K := O_K/p_K and f = f_K is the degree of K over the finite field with p elementsFp. Forγ ∈ O_Kthe classγ+pK is denoted byγ. The ramification index of p_K is denoted by e= e_K and we recall that ef = n. By d_K we denote the discriminant of K and by dϕ the discriminant of a polyno- mialϕ.

2. Units

It is well known that the group of units of a p-adic field K can be de- composed into a direct product

K^∗=hπ_Ki × hζ_Ki ×(1 +p_K)∼=π^Z_K×K^∗×(1 +p_K),

whereζ_K ∈K a (#K−1)-th root of unity. The multiplicative group 1 +p_K is called the group of principal units ofK. Ifη∈1 +p_K is a principal unit withv_p(η−1) =λwe callλthe level of η.

A comprehensive treatment of the results presented in this section can be found in [Has80, chapter 15].

Lemma 2.1 (p-th power rule). Let α be in O_K. Let p = −π_K^eKε be the factorization ofpwhereεis a unit. Then thep–th power of1+απ^λ_K satisfies

(1 +απ^λ)^p ≡







1 +α^pπ^pλ_K mod p^pλ+1_K if 16λ < _p−1^eK , 1 + (α^p−εα)π_K^pλ mod p^pλ+1_K if λ= _p−1^eK , 1−εαπ_K^λ+e mod p^λ+e+1_K if λ > _p−1^eK . The maps h₁ : α+p 7−→ α^p +p_K and h₃ : α+p_K 7−→ −εα+p_K are automorphisms ofK⁺, whereas h2 :α+p_K7−→α^p−εα+p_K is in general only a homomorphism. The kernel ofh₂ is of order 1 or p.

As (1 +p^λ_K)/(1 +p^λ+1_K )∼=p^λ_K/p^λ+1_K ∼=K⁺, it follows that ifη_λ,1, . . . , η_λ,fK is a system of generators for the levelλ < _p−1^eK (for the levelλ > _p−1^eK ), then η^p_λ,1, . . . , η_λ,f^p is a system of generators for the levelpλ(for the levelλ+e_K).

If (p−1)|eK the levels based on the level λ= _p−1^eK need to be discussed separately.

We define the set of fundamental levels F_K :=

λ|0< λ < ^pe_p−1^K, p-λ .

All levels can be obtained from the fundamental levels via the substitutions presented above. The cardinality ofF_K is

#F_K =j

pe p−1

k−j

pe p(p−1)

k

=e+j

e p−1

k−j

e p−1

k

=e.

If K does not contain the p-th roots of unity then principal units of the fundamental levels generate the group of principal units:

Theorem 2.2 (Basis of 1 +p_K,µp 6⊂K). Let ω1, . . . , ω_f ∈ O_K be a fixed set of representatives of an Fp-basis of K. Ifp−1 does not divide e_K or if h2 is an isomorphism, that is, K does not contain the p-th roots of unity, then the elements

η_λ,i:= 1 +ω_iπ^λ where λ∈F_K,16i6f_K are a basis of the group of principal units 1 +p_K.

IfK contains thep-th roots of unity we need one additional generator:

Theorem 2.3 (Generators of 1 +p_K, µp ⊂ K). Assume that (p−1) | e_K and h₂ is not an isomorphism, that is, K contains the p-th roots of unity. Choose e0 and µ0 such that p does not divide e0 and such that e_K =p^µ0−1(p−1)e₀. Let ω₁, . . . , ω_f ∈ O_K be a fixed set of representatives of a Fp-basis of K with ω1 chosen such that ω^p₁^µ0 −εω^p₁^µ0−1 ≡ 0 modp_K and ω₁ 6≡0 modp_K. Choose ω∗ ∈ O_K such that x^p−εx≡ω∗modp_K has no solution. Then the group of principal units 1 +p_K is generated by

η∗:= 1 +ω∗π_K^pµ0e0 and η_λ,i := 1 +ω_iπ_K^λ where λ∈F_K,16i6f_K.

Algorithms for the computation of the multiplicative group of residue class rings of global fields and the discrete logarithm therein are presented in [Coh99] and [HPP03]. They can be easily modified for the computation of the unit group of ap-adic field modulo a suitable power of the maximal idealp.

3. Class Fields

We give a short survey over local class field theory (see [Ser63] or [Iwa86]).

Yamamoto [Yam58] proofs the isomorphy and the ordering and uniqueness theorems of local class field theory in a constructive way. He does not show that there is a canonical isomorphism.

Theorem 3.1 (Isomorphy). Let L/K be an abelian extension, then there is a canonical isomorphism

K^∗/N_L/K(L^∗)∼= Gal(L/K).

Theorem 3.2 (Ordering and Uniqueness). If L1/K andL2/K are abelian extensions, then

N_(L1∩L2)/K((L₁∩L₂)^∗) = N_L1/K(L^∗₁)N_L2/K(L^∗₂) and

N_(L1L2)/K((L1L2)^∗) = N_L1/K(L^∗₁)∩N_L2/K(L^∗₂).

In particular an abelian extensionL/K is uniquely determined by its norm group N_L/K(L^∗).

The latter result reduces the problem of constructing class fields to the construction of cyclic extensions whose compositum then is the class field.

The construction of tamely ramified class fields, which is well known and explicit, is given below. In order to prove the existence theorem of local class field theory, it remains to prove the existence of cyclic, totally ramified class fields of degreep^m (m∈N). We give this proof by constructing these fields (algorithm 6.1). The existence theorem for class fields of finite degree follows:

Theorem 3.3 (Existence). Let G ⊂ K^∗ be a subgroup of finite index.

There exists a finite abelian extension L/K with N_L/K(L^∗) =G.

Tamely Ramified Class Fields. An extensionL/K is called tamely ram- ified ifp -e_L/K. Tamely ramified extensions are very well understood. It is well known that the results of local class field theory can be formulated explicitly for this case.

Letq= #K. If Gis a subgroup of K^∗ with 1 +p_K ⊂Gthen G=hπ_K^Fζ_K^S, ζ_K^Ei ×(1 +p_K)

for some integersE |q−1,F, andS. There exists a unique tamely ramified extensionL/K with N_L/K(L^∗) =G,e_L/K =E, and f_L/K =F.

Denote byT the inertia field ofL/K. There exists a primitive (q^F−1)-th root of unity ζ_L∈L, a prime elementπ_L of L and automorphisms σ,τ in Gal(L/K) such that

• N_{T /K}(ζ_L) =ζ_K and N_L/T(π_L) =ζ_L^Sπ_K where 06t6e−1,

• ζ_L^σ =ζ_L^q and π_L^σ−1 ≡ζ

q−1 e S

L modp_L,

• ζ_L^τ =ζ_L andπ^τ−1_L =ζ

q−1 e

K .

The Galois group of L/K is generated by σ andτ:

Gal(L/K) =hσ, τi ∼=hs, t|st=ts, s^F =t^−S, t^E = idi.

The Galois group Gal(L/K) is isomorphic toK^∗/N_L/K(L^∗) by the map:

π_K 7→σ, ζ_K 7→τ, η 7→id for allη ∈1 +p_K.

Wildly Ramified Class Fields. We have seen above that subgroups of hπ_Ki correspond to unramified extensions and that subgroups of hζ_Ki correspond to tamely ramified extensions. Subgroups of K^∗ that do not contain all of 1 +p_K correspond to wildly ramified extensions.

Lemma 3.4. LetL/K be an abelian and wildly ramified extension, that is, [L:K] =p^m for some m∈N. Then

K^∗/N_L/K(L^∗)∼= (1 +p_K)/N_L/K(1 +p_L).

4. Generating Polynomials of Ramified Extensions of Degree p LetK be an extension ofQp of degreen=ef with ramification indexe, prime ideal p, and inertia degreef. Setq :=p^f = #K. For α, β ∈ O_K we write α≡β ifν_K(α−β)> ν_K(α).

In this section we present a canonical set of polynomials that generate all extensions of K of degree p. These were first determined by Amano [Ama71] using different methods. MacKenzie and Whaples [MW56, FV93]

use p-adic Artin-Schreier polynomials in their description of extensions of degree p.

There are formulas [Kra66, PR01] for the number of extensions of a p-adic field of a given degree and discriminant given by:

Theorem 4.1 (Krasner). Let K be a finite extension of Qp, and let j = aN+b, where 06b < N, be an integer satisfying Ore’s conditions:

min{v_p(b)N, v_p(N)N}6j6v_p(N)N.

Then the number of totally ramified extensions of K of degree N and dis- criminant p^N+j−1 is

#KN,j =









 n q

ba/ec

P

i=1

eN/pⁱ

if b= 0, and n(q−1)q

ba/ec

P

i=1

eN/pⁱ+b(j−ba/eceN−1)/p^ba/ec+1c

if b >0.

There are no totally ramified extensions of degree N with discriminant p^N+j−1_K , if j does not satisfy Ore’s conditions.

Letj =ap+b satisfy Ore’s conditions for ramified extensions of degree pthen

#Kp,j =

pq^e ifb= 0 p(q−1)q^a ifb6= 0.

We give a set of canonical generating polynomials for every extension in K_p,j with j satisfying Ore’s conditions.

First, we recall Panayi’s root finding algorithm [Pan95, PR01] which we apply in the proofs in this section. Second, we determine a set of canonical generating polynomials for pure extensions of degree p of a p-adic field, that is, for the case b = 0. Third, we give a set of canonical generating polynomials for extensions of degree p of discriminant p^p+ap+b−1, where b6= 0, of a p-adic field.

Root finding. We use the notation from [PR01]. Let ϕ(x) = c_nxⁿ +

· · ·+c0 ∈ O_K[x]. Denote the minimum of the valuations of the coeffi- cients ofϕ(x) by ν_K(ϕ) := min

ν_K(c₀), . . . , ν_K(c_n) and define ϕ^#(x) :=

ϕ(x)/π^νK(ϕ). For α ∈ O_K, denote its representative in the residue class fieldK byα, and for β ∈K, denote a lift of β toO_K by β.b

In order to find a root ofϕ(x), we define two sequences (ϕi(x))iand (δi)i

in the following way:

• set ϕ0(x) :=ϕ^#(x) and

• let δ₀ ∈ O_K be a root of ϕ₀(x) modulo p_K. Ifϕ^#_i (x) has a root β_i then

• ϕ_i+1(x) :=ϕ^#_i (xπ+βb_i) and

• δi+1:=βbiπⁱ⁺¹+δi.

If indeed ϕ(x) has a root (in O_K) congruent to β modulo p, then δ_i is congruent to this root modulo increasing powers of p. At some point, one of the following cases must occur:

(a) deg(ϕ^#_i ) = 1 and δi−1 is an approximation of one root of ϕ(x).

(b) deg(ϕ^#_i ) = 0 and δi−1 is not an approximation of a root ofϕ(x).

(c) ϕ^#_i has no roots and thus δi−1 is not an approximation of a root of the polynomial ϕ(x).

While constructing this sequence it may happen that ϕ_i(x) has more than one root. In this case we split the sequence and consider one sequence for each root. One shows that the algorithm terminates with either (a), (b), or (c) after at mostνK(dϕ) iterations.

Extensions of p-adic fields of discriminant p^p+pe−1. Let ζ be a (q−1)-th root of unity and set R= (ρ₀, . . . , ρq−1) = (0,1, ζ, ζ², . . . , ζ^q−2).

The setRis a multiplicative system of representatives of K inK.

Theorem 4.2. Let J :=

r ∈ Z | 1 6 r < pe/(p−1), p - r . Each extension of degree p of K of discriminant p^p+ep−1 is generated by a root of exactly one of the polynomials of the form

ϕ(x) =















x^p+π+X

i∈J

ρciπⁱ⁺¹+kδπ^pe/(p−1)+1 if







(p−1)|e and x^p−1+ (p/π^e) is reducible, x^p+π+X

i∈J

ρciπⁱ⁺¹ otherwise,

where δ ∈ O_K is chosen such that x^p −x+δ is irreducible over K and 0 6 k < p. These extensions are Galois if and only if (p−1) | e and x^p−1+p/π^e is reducible, i.e., if K contains the p-th roots of unity.

It is obvious that a pure extension can be Galois only ifK contains the p-th roots of unity. We prepare for the proof with some auxiliary results.

Lemma 4.3. Assume that ϕ(x) := x^p−1 +c ∈ Fq[x] has p−1 roots in Fq. Then there exists d ∈ Fq such that ψk(x) := x^p +cx−kd∈ Fq[x] is irreducible for all 16k < p.

Proof. Leth(x) =x^p+cx∈Fq[x]. Asϕ(x) splits completely overFq, there exists d∈ Fq\h(Fq). Now ψ1(x) = x^p +cx−d is irreducible. It follows that

kψ1(x) =kx^p+ckx−kd= (kx)^p+c(kx)−kd

is irreducible. Replacing kx by y we find that ψ_k(y) = y^p +cy −kd is

irreducible overFq.

Lemma 4.4. Let

ϕ_t(x) =x^p+π+X

r∈J

ρ_ct,rπ^r+1+k_tδπ^v+1∈ O_K[x] (t∈ {1,2}) where ρct,r ∈ R, v >pe/(p−1), and δ ∈ O_K. Let α1 be a zero of ϕ1 and α2 be a zero of ϕ2 in an algebraic closure of K.

(a) If c_1,r 6=c_2,r for some r∈J, then K(α₁)K(α₂).

(b) If c1,r =c2,r for all r ∈J, if K contains the p-th roots of unity, δ is chosen such thatx^p−x+δ is irreducible,v=pe/(p−1), andk₁6=k₂ then K(α1)K(α2).

Proof. Let L1:=K(α1) and letp₁ denote the maximal ideal of L1.

(a) We use Panayi’s root-finding algorithm to show that ϕ₂(x) does not have any roots over K(α1). As ϕ2(x) ≡ x^p mod (π), we set ϕ2,1(x) :=

ϕ₂(α₁x). Then

ϕ_2,1(x) =α^p₁x^p+π+X

r∈J

ρ_c2,rπ^r+1+k₂δπ^v+1

=

−π−X

r∈J

ρc1,rπ^r+1−k1δπ^v+1

x^p+π+X

r∈J

ρc2,rπ^r+1+k2δπ^v+1

≡π(−x^p+ 1).

Henceϕ^#_2,1(x) =ϕ_2,1(x)/π≡ −x^p+ 1 and we set ϕ_2,2(x) :=ϕ^#_2,1(α₁x+ 1)

=

−1−X

r∈J

ρ_c1,rπ^r−k₁δπ^v

(α₁x+ 1)^p+ 1 +X

r∈J

ρ_c2,rπ^r+k₂δπ^v

≡

−1−X

r∈J

ρc1,rπ^r−k1δπ^v

α^p₁x^p+ 1 +X

r∈J

ρc2,rπ^r+k2δπ^v. Let β_i be a root of ϕ^#_2,i+2. Let m be minimal with c_1,m = c_2,m. Then βm 6= 0. Letm < u < pe/(p−1). Assume that the root-finding algorithm does not terminate with degϕ^#_2,w = 0 for some m 6 w 6 u. After u iterations of the root-finding algorithm, we have

ϕ_2,u+1(x) =

−1−X

r∈J

ρ_c1,r+1π^r−k₁δπ^v

· (α^u₁x+βu−1α^u−1₁ +· · ·+βmα^m₁ + 1)^p + 1 +X

r∈J

ρc2,r+1πⁱ+ +k2δπ^v−1

≡ −α^pu₁ x^p−pα^u₁x−pβmα^m₁ + X

r∈J ,r>m

(ρc2,r+1−ρc1,r+1)π^r. The minimal valuation of the coefficients ofϕ_2,u+1(x) is eitherν_L1(α^pu₁ ) = puorνL1(pβmα^m₁ ) =pe+m. As gcd(p, m) = 1 andm < pe/(p−1), there existsu∈Nsuch that the polynomialϕ^#_2,u+1(x) is constant. Thus the root- finding algorithm terminates with the conclusion that ϕ2(x) is irreducible overK(α₁).

(b) We set ϕ_2,1(x) := ϕ₂(α₁x) and ϕ_2,2(x) := ϕ^#₁ (α₁x+ 1). After v +

1 iterations of the root-finding algorithm we obtain ϕ2,v+2(x) ≡ −α^vp₁ x^p

−pα₁^vx+ (k2 −k1)δπ^v. By lemma 4.3 ϕ^#_2+v(x) is irreducible for k1 6= k2. Therefore,ϕ₂(x) has no root inK(α₁) and ϕ₁(x) and ϕ₂(x) generate non-

isomorphic extensions overK.

Proof of theorem 4.2. We will show that the number of extensions given by the polynomialsϕ(x) is greater then or equal to the number of extensions given by theorem 4.1. The number of elements inJ ise(see section 2).

By lemma 4.4 (a), the roots of two polynomials generate non-isomorphic extensions if the coefficients ρci differ for at least one i ∈ J. For every i we have the choice amongp^f =q values for ρci. This gives q^e polynomials generating non-isomorphic extensions.

IfKdoes not contain thep-th roots of unity, then an extension generated by a root α of a polynomial ϕ(x) does not contain any of the other roots of ϕ(x). Hence the roots of each polynomial give p distinct extensions of K. Thus our set of polynomials generates allpq^e extensions.

IfK contains the p-th roots of unity, then lemma 4.4 (b) gives us p−1 additional extensions for each of the polynomials from lemma 4.4 (a). Thus our set of polynomials generates allpq^e extensions.

Extensions of p-adic fields of discriminant p^p+ap+b−1, b6= 0.

Theorem 4.5. Let J :=

r ∈ Z | 1 6 r < (ap+b)/(p−1), p - (b+r) and if(p−1)|(a+b), setv= (ap+b)/(p−1). Each extension of degreep of K of discriminant p^p+ap+b−1 withb6= 0 is generated by a root of exactly one of the polynomials of the form

ϕ(x) =















x^p+ζ^sπ^a+1x^b+π+X

i∈J

ρ_ciπⁱ⁺¹+kδπ^v+1 if







(p−1)|(a+b) and x^p−1+ (−1)^ap+1ζ^sb has p−1 roots , x^p+ζ^sπ^a+1x^b+π+X

i∈J

ρciπⁱ⁺¹ otherwise,

where ρ ∈ R and δ ∈ O_K is chosen such that x^p + (−1)^ap+1ζ^sbx+δ is irreducible inK and 06k < p. These extensions are Galois if and only if (p−1)|(a+b) and x^p−1−ζ^sb∈K[x] is reducible.

Lemma 4.6. Let

x^p+ζ^stπ^a+1x^b+π+X

r∈J

ρ_ct,rπ^r+1+k_tδ_tπ^v+1∈ O_K[x] (t∈ {1,2}) where ρt,r ∈ R, v > ^ap+b_p−1 , and δt ∈ O_K. Let α1 be a zero of ϕ1 and α2 be a zero of ϕ2 in an algebraic closure of K.

(a) If s₁ 6=s₂, then K(α₁)K(α₂).

(b) If s1 =s2 andc1,r 6=c2,r for some r∈J thenK(α1)K(α2).

(c) K(α1)/K is Galois if and only if a+b≡0 mod (p−1) and x^p−1+ (−1)^ap+1ζ^s1b is reducible over K.

(d) Assume s1 = s2 and c1,r = c2,r for all r ∈ J. If (p−1) |(ap+b), then for v = ^ap+b_p−a there exists δ ∈ O_K such that K(α1) K(α2) if k16=k2.

Proof. Let L1:=K(α1).

(a) Fort∈ {1,2}letγt=P

r∈Jρct,rπ^r+ktδtπ^v. Thenα^p₁/π=−ζ^s1π^aα^b− 1−γ₁. We use Panayi’s root-finding algorithm to show thatϕ₂(x) has no root overL₁ =K(α₁). As before, we getϕ_2,1(x) :=ϕ₂(α₁x)≡π(−x^p+ 1).

Therefore we set

ϕ2,2(x) :=ϕ^#_2,1(α1x+ 1)

= (−ζ^s1π^aα^b−1−γ₁)(α₁x+ 1)^p+ζ^s2π^aα^b(α₁x+ 1)^b+ 1 +γ₂.

Let 2 6 u 6 pe/(p−1). Let β_i ∈ R be a root of ϕ^#_2,i(x). Assume that the root-finding algorithm does not terminate with degϕ^#_2,w = 0 for some 2 6 w 6 u and let m be minimal with m < u < pe/(p−1) and β_m 6≡

0 mod (α). Afteru iterations of the root-finding algorithm, we have ϕ2,u+1(x) = (−ζ^s1π^aα^b₁−1−γ1)(α^u₁x+βu−1α^u−1₁ +· · ·+βmα^m₁ + 1)^p

+ζ^s2tπ^aα^b(α^u₁x+βu−1α^u−1₁ +· · ·+β_mα^m₁ + 1)^b+ 1 +γ₂π.

Because u 6 e, νL1(p) = pe, and a < e, the minimal valuation of the coefficients of ϕ2,u+1(x) is either νL1(−α^pu₁ ) = pu or νL1(π^aα^b₁) = pa+ b. Hence the root-finding algorithm terminates with ϕ_2,u+1(x) ≡ (ζ^s2 − ζ^s1)π^aα^b for someu in the range 26u6e.

(b) We show that ϕ2(x) does not have any roots over L1. As ϕ2(x) ≡ x^p mod (π), we getϕ_2,1(x) :=ϕ₂(αx). Now ϕ^#_2,1(x) ≡ −x^p+ 1 and we set ϕ_2,2(x) :=ϕ^#_2,1(α₁x+ 1).

Denote by βr a root of ϕ^#_2,r+1(x). Let m be minimal with m < u <

pe/(p−1) and β_m 6≡ 0 mod (α). Assume that the root-finding algorithm does not terminate earlier with degϕ^#_2,w = 0 for some w 6 u. After u

iterations, we have ϕ2,u+1(x) =

−ζ^s1π^aα^b₁−1−X

r∈J

ρc1,r+1πⁱ−ρc1,a+2π^a+1

·(α^u₁x+βu−1α^u−1₁ +· · ·+β_mα^m₁ + 1)^p

+ζ^s1π^aα^b₁(α^ux+βu−1α^u−1₁ +· · ·+β_mα^m₁ + 1)^b+ 1

+X

r∈J

ρ_c2,r+1π^r+ρ_c2,a+1π^a+1

≡ −α^pux^p−pα^u₁x−pβ_mα^m₁ −X

r∈J

ρ_c1,r+1π^r(β_mα^m₁ )^p−(β_mα^m₁ )^p +ζ^s1π^aα^b₁bα^u₁x+ζ^s1π^aα^b₁bβmα^m+X

r∈J

(ρc2,r+1−ρc1,r+1)π^r, withβ_m 6≡0 mod (α₁).

The minimal valuation of the terms ofϕ2,u+1(x) is νL1(ζ^s1π^aα^b₁bβmα^m₁ ) =pa+b+m

orν_L1(α^pr₁ ) =pr. By the choice ofJwe havep-(pa+b+m). Therefore, the root-finding algorithm terminates with ϕ2,u(x) ≡ ζ^sπ^aα^b₁bβmα^m for some u∈N.

(c) We show that ϕ1(x) splits completely over L1 if and only if the con- ditions above are fulfilled. We set ϕ_1,1(x) := ϕ₁(α₁x) and ϕ_1,2(x) :=

ϕ^#_1,1(αx+ 1). Thus

ϕ1,2(x) = −ζ^s1π^aα^b₁−1−P

r∈Jρc1,rπ^r

(α1x+ 1)^p +ζ^s1π^aα^b(α1x+ 1)^b+ 1 +P

r∈Jρc1,rπ^r

≡x(−α^p₁x^p−1+ζ^s1π^aα^b+1₁ b).

After u+ 1 iterations we get

ϕ1,u+1(x)≡











−α^up₁ x^p ifup < pa+b+u, x(−α^up₁ x^p−1+ζ^s1π^aα^b+u₁ b) ifup=pa+b+u, ζ^s1π^aα^b+1₁ bx if

up > pa+b+u and (p−1)-(a+b).

In the third case,ϕ^#_1,u+1(x) is linear and therefore ϕ1(x) has only one root overL1. In the second case,

ϕ_u+1(x)≡ −α^up₁ x^p+ζ^s1π^aα^b+u₁ bx≡ −α^up₁ x^p+ζ^s1(−α₁)^apα^b+ux

and so ϕ^#_1,u+1(x) ≡ −x^p+ (−1)^apζ^s1bxmod (α₁). If ϕ^#_u+1(x) has p roots overK for every root β of ϕ^#_1,u+1(x), we get

ϕ_1,u+2(x) =ϕ_1,u+1(α₁x+β)

≡ −α^(u+1)p₁ x^p+ (−1)^apα^u+1₁ βζ^s1π^aα^b₁+ (−1)^apα^u+1₁ bβ^bζ^s1π^aα^b₁x.

Butup+p > u+ 1 +pa+b; thusϕ^#_1,u+2(x) is linear andϕ₁(x) has as many distinct roots as ϕ^#_1,u+1(x).

(d) We set ϕ_2,1(x) := ϕ(αx) and ϕ_2,2(x) := ϕ^#_2,1(αx+ 1). We obtain ϕ2,v+1(x)≡ −α^vp₁ x^p+ζ^s1π^aα^b+v₁ bx+ (k1−k2)δπ^v,hence ϕ^#_v+1(x) = x^p + (−1)^ap+1ζ^s1bx+ (k₁−k₂)δ. By lemma 4.3, there exists δ ∈ O_K such that

ϕ^#_2,v+1(x) is irreducible.

Proof of theorem 4.5. If (p−1)-(a+b), then

#J =a+ ja+b

p−1

k

−j

a+b

p +_p(p−1)^a+b k

−j

b p

k

=a+j

a+b−1 p−1

k−ja(p−1)+a+b(p−1)+b p(p−1)

k

=a.

If (p−1)|(a+b), then

#J =a+^a+b_p−1−1−j

a+b

p +_p(p−1)^a+b −1k

−j

b p

k

=a+^a+b−1_p−1 −j

a+b−1 p−1

k

=a.

Using lemma 4.6 (a), we getp^f−1 sets of generating polynomials. By lemma 4.6 (b), each of these sets contains p^{f a} polynomials that generate non- isomorphic fields. Now either the roots of one of the polynomials generate pdistinct extensions or the extension generated by any root is cyclic. In the latter case, we havep−1 additional polynomials generating one extension each by lemma 4.6 (d). Thus we obtain (p^f −1)p^af+1 distinct extensions.

Number of Galois Extensions. The following result can also easily be deduced from class field theory.

Corollary 4.7. Let K be an extension of Qp of degree n. If K does not contain the p-th roots of unity, then the number of ramified Galois exten- sions ofK of degreep isp·^p_p−1ⁿ⁻¹.If K contains thep-th roots of unity then the number of ramified Galois extensions of K of degree p isp·^pn+1_p−1⁻¹. Proof. Letϕ(x) be as in theorem 4.5. We denote the inertia degree and the ramification index of K by f and e, respectively. The number of values of sfor whichx^p−1−ζ^s is reducible is (p^f −1)/(p−1). By Ore’s conditions, 06a6e. For everya < e, there is exactly onebwith 16b < p such that (p−1) | (a+b). For every a, the set J contains a elements. This gives

p^{f a} combinations of values of ci,i∈J. We have pchoices for k. Thus the number of polynomialsϕ(x) generating Galois extensions is

p·p^f −1 p−1 ·

e−1

X

a=0

p^{f a}=p·p^f −1

p−1 ·p^{f e}−1

p^f −1 =p·pⁿ−1 p−1 .

IfK contains the p-the roots of unity, a=e also yields Galois extensions.

By theorem 4.2, we obtain additionalp(p^f)^e=pⁿ⁺¹ extensions.

5. Ramified Abelian Extensions of Degree p

LetL/K be an abelian ramified extension of degreep. The ramification number (Verzweigungszahl) ofL/K is defined asv=v_L/K =ν_L(π_L^σ−1−1), whereσ ∈Gal(L/K)\ {id}. The ramification numberv is independent of the choice ofσ. Ifϕis the minimal polynomial of πL, then

νL(d(ϕ)) =X

i6=j

νL σⁱ(πL)−σ^j(πL)

=

p(p−1)

X

i=1

νL σ(πL)−πL

= p(p−1)(v+ 1).

Hence,ν_K(d_L/K) = (p−1)(v+ 1) for the discriminant ofL/K andD_L/K = p^(p−1)(v+1)_L for the different of the extension. It follows from Ore’s conditions (see Theorem 4.1) that eitherv=p_p−1^eK orv= ^ap+b_p−1 ∈FK wherej=ap+b satisfies Ore’s conditions.

Lemma 5.1. LetL/K be a ramified extension of degreep. Ifd:=ν_L(D_L/K)

= (p−1)(v+ 1), then

T_L/K(p^m_L) =p _m+d

eL/K

K .

See [FV93, section 1.4] for a proof. We use Newton’s relations to inves- tigate the norm group of abelian extensions of degreep.

Proposition 5.2 (Newton’s relations). Let ϑ=ϑ⁽¹⁾, . . . , ϑ⁽ⁿ⁾ be the roots of a monic polynomial ϕ = P

06i6nγ_ixⁱ. Then γ_i = (−1)⁽ⁿ⁻ⁱ⁾Rn−i(ϑ) where Rn−i(ϑ) is the (n−i)-th symmetric function in ϑ⁽¹⁾, . . . , ϑ⁽ⁿ⁾. Set Sk(ϑ) =Pn

i=1 ϑ⁽ⁱ⁾k

for each integer k>1. Then S_k(ϑ) =

( −kγ_n−k−Pk−1

i=1 γn−iSk−i(ϑ) for 16k6n, and

−Pn

i=1γn−iSk−i(ϑ) for k > n.

The following describes explicitly where and how the jump in the norm group takes place (c.f. [FV93, section 1.5]).

Lemma 5.3. Let L/K be ramified abelian of degree p and let v denote the ramification number of L/K. Let hσi = Gal(L/K). Assume that N_L/K(πL) =πK. Letε∈K be chosen such thatπ_L^σ−1 ≡1 +επ^v_Lmodp^v+1. Then

N_L/K(1 +απ_Lⁱ)≡1 +α^pπ_Kⁱ mod pⁱ⁺¹_K ifi < v, N_L/K(1 +απ_L^v)≡1 + (α^p−ε^p−1α)π_K^v mod p^v+1_K , and N_L/K(1 +απ_L^v+p(i−v))≡1−ε^p−1απ_Kⁱ mod pⁱ⁺¹_K ifi > v.

The kernel of the endomorphism K⁺→K⁺ given by α7→α^p−ε^p−1α has order p.

Proof. We have

N_L/K(1 +ωπ_Lⁱ) = 1 +ωR₁(πⁱ_L) +ω²R₂(πⁱ_L) +· · ·+ω^pR_p(πⁱ_L), whereR_k(π_Lⁱ) denotes thek-th symmetric polynomial inπⁱ_L, π^σi_L, . . . , π^σ_L^p−1i. In particularR₁(π_Lⁱ) = T_L/K(πⁱ_L) andR_p(π_Lⁱ) = N_L/K(π_L)ⁱ. By lemma 5.1 and ν_L(D_L/K) = (v+ 1)(p−1), we obtain

S_k(π_Lⁱ) = T_L/K(π_L^ki)∈T_L/K(p^ki_L)⊂p^λ_K^ki where

λ_ki=(p−1)(v+1)+ki p

=v+ 1 +_−v−1+ki

p

=v+_ki−v

p

=v−_v−ki

p

. (i) If i < v, then i < λ1 = v−_v−i

p

and νK(S_k(π_Lⁱ)) > λ_k > λ1 > i.

With Newton’s relations we getν_K(R_k(πⁱ_L))> i for 16k 6p−1 and as R_p(π_Lⁱ) = N_L/K(π_L)ⁱ=πⁱ_K, we obtain

N_L/K(1 +απ_Lⁱ)≡1 +α^pπ_Kⁱ mod pⁱ⁺¹_K .

(ii) Assume i = v. By lemma 5.1 T_L/K(p^v_L) = p^λ_K^v, and so T_L/K(π_L^v) ≡ βπ_K^v mod p^v+1_K for some β ∈ O_K^∗. We have λ_k = v+(k−1)v

p

> v. If k > 2 then νK(Sk(πⁱ_L)) > λk > v + 1. Hence with Newton’s relations νK(Rk(π_Lⁱ))>min(kv, v+ 1) >v+ 1 for 26k 6p−1. Thus N_L/K(1 + απ^v_L)≡1 +αβπ^v_K+α^pπ^v_K mod p^v+1_K and N_L/K(1 +π_L^v)⊂(1 +p_K)^v. By the definition of ε and as N_L/K(π^σ−1_L ) = 1 we have N_L/K(1 +επ_L^v) ≡ 1 mod p^v+1_K . Thereforeβ ≡ −ε^p−1 mod p_K and we conclude that

N_L/K(1 +απ_L^v)≡1 + (α^p−ε^p−1α)π_K^v mod p^v+1_K .

(iii) Let i > v. We have λ_v+p(i−v) = i and λk(v+p(i−v)) > i. By the considerations in (ii), we obtain

N_L/K(1 +απ^v+p(i−v)_L )≡1−ε^p−1απⁱ_K mod pⁱ⁺¹_K .

Next we investigate the relationship between the minimal polynomial of π_L, a uniformizer of the field L, and the norm group N_L/K(L^∗). We start by choosing a suitable representation for subgroups ofK^∗ of index p. We begin with extensions of discriminant p^p+eL/K−1, which are Galois if and only if K contains the p-the roots of unity.

IfK contains thep-th roots of unity then

K^∗ =hζ_Ki × hπ_Ki × hη_λ,i |λ∈FK,16i6fK;η∗i

Let G be a subgroup of K^∗ of index p with η∗ ∈/ G. Let (g₁, ..., g_eKfK+3) be generators ofG. Letn=eKfK and B ∈Z^n+3×n+3 such that

(g₁, ..., g_eKfK+3)^T =B(ζ_K, π_K, η_λ,i|λ∈F_K,16i6f_K, η∗)^T be a representation matrix of G. Let A be the row Hermite normal form ofB. Then

A=



























1 0 · · · 0 aπ

0 1 0 0 0

... 0 1 0 · · · 0 a1,1

... . .. ... ... ... ... ... . .. ... 0 ...

... . .. 1 av−1,f

0 · · · 0 p

























 .

Thus G=

π_Kη^a_∗^π; ζ_K; η_λ,iη^a∗^λ,i

λ∈F_K,16i6f_K; η^p_∗

(t∈ {1,2}).

Theorem 5.4. Assume that K contains the p-th roots of unity. Let ϕ_t(x) =x^p+π+X

r∈J

ρ_ct,rπ^r+1+k_tδπ^v+1∈ O_K[x] (t∈ {1,2}) be polynomials as in theorem 4.2. IfL1 :=K[x]/(ϕ1) andL2:=K[x]/(ϕ2), thenv=v_L1/K =v_L2/K =pe_K/(p−1). Hence

N(L^∗_t) =

πKη∗^at,π; ζK; ηλ,iη∗^at,λ,i

λ∈FK,16i6fK; η_∗^p

(t∈ {1,2}).

(a) Let w ∈ J = {1 6 r 6 pe/(p−1) | p - r}. We have c_1,r = c_2,r for 16r < w, r∈J if and only if a1,v−r,i =a2,v−r,i for all 16r < w, r ∈J and all 16i6f_K.

(b) If c1,r =c2,r for all r∈J, then k1 =k2 if and only if a1,π =a2,π. Proof. (a)We show one implication directly. The other implication follows by a counting argument.

(i) Asp|(v−λ) if and only ifp|λwe havev−r∈F_K if and only ifr ∈J. (ii) Let π_t be a root of ϕ_t. We write ϕ_t = x^p −γ_t. The minimal poly- nomial of π_t^λ over K is x^p−γ_t^λ. The characteristic polynomial of ωπ_t^λ is

x^p+ω^pγ_t^λ. The characteristic polynomial of 1+ωπ^λ_t is (x−1)^p−α^pω^λ. Thus N_Lt/K(1 +ωπ_t^λ) = (−1)^p −ω^pγ_t^λ. If γ₁ =γ₂+απ^w+1_K for some α ∈ O_K^∗ then forr6wwe obtain

γ₁^v−r = (γ₂+π_K^w+1α)^v−r≡γ₂^v−r+ (v−r)γ₂^v−r−1απ_K^w+1 mod p^v+1_K .

(iii) Assume that c_1,r =c_2,r for all 16r < w. For all r6w−1 we obtain N_L1/K(1 +ωπ_L^v−r

1 ) = (−1)^p+ω^pγ₁^v−r≡N_L2/K(1 +ωπ_L^v−r

1 ) modp^v+1_K which impliesa1,v−r,i =a2,v−r,i for all 16r < w and 16i6fK. (iv) Ifc_1,w6=c_2,w forr=wwe have

N_L2/K(1 +ωπ^v−w_L

2 ) = (−1)^p+ω^pγ₂^v−w and

N_L1/K(1 +ωπ^v−w_L

1 )≡(−1)^p+ω^p(γ₂^v−w+ (v−w)γ₂^v−w−1π^w+1_K α) mod p^v+1_K . By (i),p-(v−w) and asν_K(γ₂) = 1, it follows that a_1,w,i6=a_2,w,i.

(v) There arep^f choices for eachρct,r. On the corresponding levelλ=v−r there aref generating principal units η_λ,1, . . . , η_λ,1 with in total p^f choices for the exponentsa_t,λ,1, . . . , a_t,λ,f. This shows the equivalence.

(b)We have

N_Lt/K(πLt)≡πK+X

r∈J

ρct,rπ^r+1_K +ktδπ_K^v+1 ≡πK

Y

λ,i

η_λ,i^at,λ,i·η^a∗^t,π

!

modp^v+2_K .

Since c_1,r = c_2,r for all r ∈ J, we have a_1,λ,i = a_2,λ,i for all λ ∈ F_Lt, 16i6f. Thusk1=k2 is equivalent toa1,π=a2,π.

IfK does not contain thep-th roots of unity then K^∗ =hπ_Ki × hζ_Ki × Y

λ∈F_K

Y

16i6fK

hη_λ,ii.

Let G be a subgroup of K^∗ of index p and let A be the row Hermite normal form of the representation matrix ofG. There are valuesλ₀∈F_K,