EXTENSIONS WITH ALMOST

MAXIMAL DEPTH OF RAMIFICATION

S. V. Vostokov, I. B. Zhukov

November 8, 1999

Introduction

§1 of [BVZ] was devoted to a general discussion of the extensions L/K which satisfy the condition

(*)  [L : K] | D_{L/K}.

Here L/K is a wildly ramified Galois extension of complete discretely valued fields of characteristic 0 with the residue field of prime characteristic p, D_{L/K} is the different of L/K. The importance of the condition (*) is clarified in [V, BVZ, BLV]. Roughly speaking, it appeared that (*) is equivalent to the fact that some ideals of O_{L} have a non-trivial decomposition into a direct sum of O_{K}[G]-modules, where O_{K} is the ring of integers in K, G = Gal(L/K).

The condition (*) means that the depth of ramification in L/K has its “almost maximal” value. For example, consider the case [L : K] = p. Then we have v_{L}(D_{L/K}) = (p − 1)(h + 1) = pd + p − 1, where h is the ramification jump of L/K, and d = d_{K}(L/K) is the depth of ramification. The condition (*) can be read here as pe_{K} ⩽ (p − 1)(h + 1) which is equivalent to h ⩾ \frac{pe_{K}}{p−1} − 1, or to d ⩾ e_{K} − \frac{p−1}{p}. Recall that the maximal possible values for h and d are \frac{pe_{K}}{p−1} and e_{K} respectively.

Consider the ramification filtration on Gal(L/K). It yields a canonical decomposition of L/K into a tower of elementary abelian p-extensions. We can further decompose L/K into a tower of cyclic extensions of degree p: K = K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{n} = L. The following observations were reflected in [BVZ].

A. The condition (*) for L/K implies (*) for all intermediate extensions in L/K. (See [BVZ, Prop. 1.5].)

B. The condition (*) for the “upper level” L/K_{n−1} implies (*) for the whole L/K. (See [ibid].)

C. The condition (*) for the “lower level” K_{1}/K does not imply (*) for the whole L/K. However, if the depth of ramification for K_{1}/K is maximal, i. e., equals e, then L/K is cyclic, and the depth of ramification is maximal on all levels. (Recall that K_{1}/K is a subextension of degree p in the subextension of L/K corresponding to the first ramification jump.) In particular, (*) for L/K holds. (This is well known, see, e. g., [Se, Ch. IV, §2, Ex. 3f].)

Typeset by AMS-TEX

Partially supported by a grant of RFBR (project No. 97-01-00058-a).

D. The condition (*) determines the Galois group of L/K, namely, forces it to be cyclic when p > 2. (See [BVZ, Prop. 1.5].)

E. If we know a priori that L/K is cyclic, then the condition (*) for the “lower level” K_{1}/K implies (*) for the whole L/K. (It follows from [BVZ, Lemma 1.2.1].)

The aim of the present article is to consider the general case without the assumption that L/K is wildly ramified. In other words, we admit here that \overline{L}/\overline{K} can be inseparable and try to find reasonable generalizations of the above A,B,C,D,E.

§§1–3 contain some preliminary results. In §4 we establish a general structure theorem for extensions satisfying (*) under the condition that p − 1|e. In §5 we give a complete classification of cyclic extensions satisfying (*). §6 and §7 contains the proofs of the following theorems, descrbing the structure of abelian AMR extensions. In the statements we use the notion of independently ramified extensions, explained in the beginning of §3.

Theorem 1. Suppose that p − 1|e. Let E/K be an abelian p-extension. Denote by E_{1}/K the maximal subextension of E/K of exponent p. Then the following conditions are equivalent.

1. E/K satisfies (*).

2. E_{1}/K satisfies (*).

3. E = MN, where M/K and N/K are independently ramified, M/K is cyclic and satisfies (*), N/K is a maximally ramified, i. e., its depth of ramification equals ne, where p^{n} = [N : K].

Theorem 2. Suppose that p − 1 ∤ e. Let L/K be an abelian p-extension. Write L as K^{1} … K^{m}, where K^{i}/K are linearly disjoint cyclic, i = 1, …, m. Then L/K satisfies (*) if and only if the following 3 conditions are satisfied.

1. The subextension of degree p in K^{i}/K satisfies (*) for all i = 1, …, m.

2. The subextension of degree p in K^{i}/K are independently ramified, i = 1, …, m.

3. ε ∑_{i=1}^{m}(1 − \frac{1}{p^{r_{i}}} ) ⩽ 1 − \frac{1}{p^{r}}, where p^{r_{i}} = [K^{i} : K], i = 1, …, m; r =

max(r_{1}, …, r_{m}); ε = \frac{pe}{p−1} − [\frac{pe−1}{p−1}].

Finally, in §8 we establish some further properties of composites of cyclic extensions satisfying (*).

Considerable part of this work was done during the stay of the second author in Max-Planck-Institut für Mathematik in Bonn. The second author expresses his gratitude to this institute for hospitality and excellent working conditions.

Notation and terminology

p is a prime number; we assume throughout the article that p > 2.

Throughout the article the letter K denotes a complete discretely valued field of characteristic 0 or p with the residue field of characteristic p. For such a field K, we use the following notation:

• v = v_{K}: the valuation on K, as well as its non-normalized extension onto the algebraic closure of K;

• O = O_{K}: the valuation ring in K;

• 𝔐 = 𝔐_{K} = { a ∈ O_{K} : v(a) > 0 }: the maximal ideal of this ring;

• 𝔭(i) = 𝔭_{K}(i) = 𝔐_{K}^{i} for any i ∈ ℤ; • U_{K} = O_{K}^{∗};

• U_{i,K} = 1 + 𝔭_{K}(i) for i ⩾ 1;

• \overline{K}: the residue field of K;

• \overline{a}: the residue class in \overline{K} of a ∈ O_{K};

• e = e_{K} = v_{K}(p): the absolute ramification index of K;

• e(L/K): the ramification index of a finite extension L/K;

• D_{L/K}: the different of a finite extension L/K;

• ζ_{m}: a primitive mth root of unity in the algebraic closure of K. A finite extension L/K is said to be:

• tame, if (p, e(L/K)) = 1;

• wild, if [L : K] = e(L/K) is a power of p;

• ferocious, if [L : K] = [\overline{L} : \overline{K}]_{insep};

• a wf-extension if the ramification subfield (i. e., the maximal tame subextension) in L/K coincides with K.

In other words, a wf-extension is an extension which can be represented by a tower of extensions of degree p, each of them is either wild or ferocious.

Let M/L be a finite separable extension of algebraic extensions of K. Depth of ramification of M/L is defined as

d_{K}(M/L) = inf\{v_{K}(\frac{Tr_{M/L} y}{y} )|y ∈ M^{∗}\}.

Let char K = 0. We say that an extension L/K is almost maximally ramified (AMR), if [L : K]|D_{L/K}. An extension L/K is maximally ramified (MR), if [L : K] = p^{s}, and d_{K}(L/K) = se.

Let L/K be a finite Galois extension, σ ∈ Gal(L/K). The ramification number of σ is defined as

h(σ) = min_{a∈L^{∗}} v_{L}(a^{σ−1} − 1).

If h(σ) is the same for all σ ≠ 1, it is called the ramification number (or ramification jump) of L/K and denoted by h(L/K). In particular, h(L/K) is always defined for a cyclic extension L/K of degree p.

§1. Properties of depth of ramifiction

The following properies are taken from the paper of Hyodo [H].

Proposition 1.1. Let L/K be a finite extension, M a finite extension of L. Then

d_{K}(M/L) = v_{K}(D_{M/L}) + v_{K}(π_{M}) − v_{K}(π_{L}).

In particular, d_{K}(M/L) = 0 if and only if M/L is tame.

Proof. Immediately from the definition of the different. □

Proposition 1.2. Let N be an intermediate field in M/L. Then

d_{K}(M/L) = d_{K}(M/N) + d_{K}(N/L).

This is Lemma (2-4) in [H].

Corollary 1.3. We have d_{K}(M(ζ_{p})/L(ζ_{p})) = d_{K}(M/L).

Proposition 1.4. Let L/K be a cyclic extension of degree p. Then d_{L}(L/K) = (p − 1)h(L/K).

This follows from Lemma (2-10) in [H].

Now we recall what are the possible values of h(L/K) in this case. For details, see [FV, Prop. III.2.3 and Ex.1 in III.2].

Proposition 1.5. 1. Let L/K be wild, h = h(L/K). Then either 0 < h < pe/(p − 1), and (p, h) = 1, or h = pe/(p − 1). The last variant is possible only when char K = 0 and ζ_{p} ∈ K.

2. Let L/K be ferocious, h = h(L/K). Then 0 < h ⩽ e/(p − 1).

Finally, we recall the Hyodo inequality, see [H, Lemma (4-1)] or [Z,§1]; see also Prop. 2.2 below.

Proposition 1.6. Let M/K be a cyclic extension of degree p^{2}, L the intermediate field. Then

d_{K}(M/L) ⩾ min((p − 1 + p^{−1})d_{K}(L/K), e − p^{−1}e + p^{−1}d_{K}(L/K)).

§2. Some calculations

Proposition 2.1. Let L/K be a cyclic extension of degree p, d = d_{K}(L/K). Then

v(\frac{N_{L/K}a}{a^{p}} − 1) ⩾ d

for any a ∈ L^{∗}.

Proof. The assertion is trivial in the unramified case, and we may assume that L/K is either wild or ferocious.

Denote v(\frac{N_{L/K}a}{a^{p}} − 1) by j(a). Since j(a^{−1}) = j(a), we have min_{a∈L^{∗}} j(a) = min_{a∈O_{L}\{0}} j(a). Next, for any a, b ∈ O_{L} we have

v(N_{L/K}(a + b) − N_{L/K}a − N_{L/K}b) ⩾ d + p min(v(a), v(b)),

and

v((a + b)^{p} − a^{p} − b^{p}) ⩾ e + p min(v(a), v(b)) ⩾ d + p min(v(a), v(b)).

Suppose that a, b ∈ O_{L}, a ≠ 0, b ≠ 0, v(a + b) = min(v(a), v(b)). Then

(1)  \begin{aligned}j(a + b) & = v(N_{L/K}(a + b) − (a + b)^{p}) − p min(v(a), v(b)) \\ & ⩾ min(j(a), j(b), d). \end{aligned}

Next, any a ∈ O_{L} can be written as ∑_{i=0}^{p−1} α_{i}x^{i}, where α_{i} ∈ O_{K}. Here x is a prime

in L, if L/K is wild, and x ∈ O_{L}, \overline{x} ∉ \overline{K}, if L/K is ferocious. It follows from (1) that

j(a) ⩾ min(j(α_{0}), j(α_{1}x), …, j(α_{p−1}x^{p−1}), d) = min(j(x), d).

Let char K = 0. Without loss of generality we may assume that ζ_{p} ∈ K. Then L = K(y), y^{p} ∈ K. If d = e, one may assume x = y, whence j(x) = ∞. Let d < e.

We may choose y so that v(y − 1) = \frac{1}{p−1}(e − d). For a suitable choice of x we have x = π^{−n}(y − 1)^{i}, where π is a prime in K, (i, p) = 1. Hence j(x) = j(y − 1). It follows from y^{p} ∈ K that

N_{L/K}(y − 1) = N_{L/K}y + (−1)^{p} = (−1)^{p−1}y^{p} + (−1)^{p}.

We have

j(y − 1) = v(\frac{y^{p} − 1}{(y − 1)^{p}} − 1) = v(y − 1) + e − pv(y − 1) = d,

and the assertion of Lemma is proved in this case.

Let now char K = p. Then L = K(y), y^{p} − y ∈ K, v(y) = −\frac{1}{p−1}d. For a suitable choice of x we have x = π^{n}y^{i}, where π, i, n are as above, and j(x) = j(y). We have N_{L/K}y = y^{p} − y, whence j(y) = v(−y^{1−p}) = d. □

Proposition 2.2. Let char K = 0, ζ_{p} ∈ K, M/K a cyclic extension of degree p^{2}, L the intermediate field, d = d_{K}(L/K). Suppose d > \frac{e}{p}.

Let L = K(x), x^{p} = 1+ρ ∈ K, v(ρ) = \frac{p}{p−1}(e−d). Then M = L(y), y^{p} = b(1+τ), b ∈ K, τ ∈ L,

v(τ^{p} − ρ) ⩾ \frac{p}{p − 1}(e − d) + min(pd − e, d).

Remark. We do not exclude the case d = e. In this case v(ρ) = v(τ) = 0.

Proof. According to Lemma 1.3 in [BVZ], we have M = L(\sqrt[p]{a}), a ∈ L, a^{σ−1} = c^{p}, c ∈ L^{∗}, N_{L/K}c ≠ 1, where σ is a generator of Gal(L/K). Let c = 1 + γ. We have

(2)  N_{L/K}c = 1 + N_{L/K}γ + Tr_{L/K} γ′,

where v(γ′ − γ) ⩾ 2v(γ). Since N_{L/K}c is a primitive root of unity of degree p, we have v(N_{L/K}γ + Tr_{L/K} γ′) = \frac{e}{p−1}. On the other hand, v(Tr_{L/K} γ′) ⩾ v(γ) + d by the definition of depth, whereas v(N_{L/K}γ) = pv(γ). Since d > \frac{e}{p}, we obtain v(γ) ⩽ \frac{e}{p(p−1)}, whence v(N_{L/K}γ) < v(Tr_{L/K} γ′), and finally v(γ) = \frac{e}{p(p−1)}.

By Proposition 2.1, v(N_{L/K}γ −γ^{p}) ⩾ \frac{e}{p−1} +d. On the other hand, v(Tr_{L/K} γ′) ⩾ \frac{e}{p(p−1)} + d. Therefore, (2) implies

N_{L/K}c ≡ 1 + γ^{p} mod 𝔭(\frac{e}{p(p − 1)} + d)O_{L},

and, furthermore,

N_{L/K}c ≡ (1 + γ)^{p} mod 𝔭(min(\frac{e}{p(p − 1)} + d, e))O_{L}.

We have (1 + γ)^{p} = c^{p} = a^{σ−1}. The element x^{σ−1} is also a primitive root of unity of degree p, whence x^{σ−1} = (N_{L/K}c)^{i} for some 1 ⩽ i ⩽ p − 1. We obtain

x^{σ−1} ≡ (a^{i})^{σ−1} mod 𝔭(min(\frac{e}{p(p − 1)} + d, e))O_{L}.

Write x as a^{i}b_{1}(1 + τ_{1}) with b_{1} ∈ K and maximal possible v(τ_{1}). We have

v(\frac{1 + τ_{1}^{σ}}{1 + τ_{1}} − 1) ⩾ min(\frac{e}{p(p − 1)} + d, e)O_{L}.

On the other hand, the choice of τ_{1} implies that τ_{1} ∉ K^{∗} ·U_{1,L}, whence v(τ_{1}^{σ−1} − 1) = \frac{d}{p−1}. Thus, v(τ_{1}) ⩾ min(\frac{e}{p(p−1)} + d, e) − \frac{d}{p−1}.

Put y = \sqrt[p]{a^{i}}. We have M = L(\sqrt[p]{a}) = L(y),

y^{p} = a^{i} = b_{1}^{−1}x(1 + τ_{1})^{−1} = b(1 + τ),

where b = b_{1}^{−1}, τ = x(1+τ_{1})^{−1}−1. To estimate v(τ^{p}−ρ), notice that x = (1+τ)(1+ τ_{1}), whence 1+ρ = x^{p} = (1+τ)^{p}(1+τ_{1})^{p}, and v(τ^{p}−ρ) ⩾ min(v(τ_{1}^{p}), v(pτ), v(pτ_{1})). Since v(x − 1) = \frac{1}{p−1}(e − d), and d > \frac{e}{p}, we have v(x − 1) < v(τ_{1}), whence v(τ) = v(x − 1) = \frac{1}{p−1}(e − d), and

v(τ^{p} − ρ) ⩾ min(v(τ_{1}^{p}), v(pτ))

= min(pv(τ_{1}), \frac{pe}{p − 1} − \frac{d}{p − 1})

= min(\frac{e}{p − 1} + pd − \frac{pd}{p − 1}, pe\frac{pd}{p − 1}, \frac{pe}{p − 1} − \frac{d}{p − 1})

= \frac{p}{p − 1}(e − d) + min(pd − e, d).

In the last equality we have used the condition p > 2. □

Remark. It is seen from the proof that v(b) = v(a^{i}), 1 ⩽ i ⩽ p − 1. On the other hand, one may assume that either p ∤ v_{L}(a), or v_{L}(a) = 0. Therefore, either p ∤ v_{L}(b), or v_{L}(b) = 0.

§3. Independently ramified extensions

Proposition 3.1. Let L/K and K′/K be any finite separable extensions, L′ = K′L. Then d_{K}(L′/K′) ⩽ d_{K}(L/K).

Proof. Let first L/K be a cyclic extension of degree p. Suppose char K = 0. In view of Corollary 1.3, we may assume ζ_{p} ∈ K. Then L = K(x), x^{p} = a ∈ K, v(a − 1) = \frac{p}{p−1}(e − d_{K}(L/K)). Since L′ = K′(x), we obtain

v(a − 1) ⩽ \frac{p}{p − 1}(e − d_{K}(L′/K′)),

whence d_{K}(L′/K′) ⩽ d_{K}(L/K). The case char K = p is analogous.

Next, let L/K be any wf-extension of degree p. Denote the normal closure of L/K by M/K and the ramification subfield in M/K by E. Then M/E is cyclic of degree p, and we have

d_{K}(L′/K′) = d_{K}(M′/K′) = d_{K}(M′/E′) ⩽ d_{K}(M/E) = d_{K}(M/K) = d_{K}(L/K).

Indeed, the extensions M/L, M′/L′, E/K, E′/K′ are tame.

The general L/K can be decomposed into a tower K ⊂ K_{1} ⊂ ⋯ ⊂ K_{n} = L, where K_{1}/K is tame, and K_{i+1}/K_{i} are wf-extensions of degree p, i = 1, …, n − 1. □

Corollary 3.2. Let L_{1}/K, L_{2}/K be finite separable extensions. Then

d_{K}(L_{1}L_{2}/K) ⩽ d_{K}(L_{1}/K) + d_{K}(L_{2}/K).

Proof. Combine Proposition 3.1 with Proposition 1.2. □

Definition. Finite separable extensions L_{1}/K, L_{2}/K, …, are said to be independently ramified (ind-ram), if d_{K}(L_{1} … L_{r}/K) = d_{K}(L_{1}/K) + ⋯ + d_{K}(L_{r}/K).

Proposition 3.3. 1. Let K_{1}/K, …, K_{r}/K be subextensions of L_{1}/K, …, L_{r}/K respectively. If L_{1}/K, …, L_{r}/K are ind-ram, then K_{1}/K, …, K_{r}/K are ind-ram as well.

2. Let r ⩾ 3. Then the extensions L_{1}/K, …, L_{r}/K are ind-ram if and only if L_{1}/K, …, L_{r−1}/K are ind-ram, and L_{r}/K is ind-ram with L_{1} … L_{r−1}/K.

Proof. Immediately from the definition. □

Remark 1. In the “classical case” (when the residue field of K is perfect) any two extensions cannot be ind-ram unless one of them is tame. Indeed, it is easy to observe that we have strict inequalities in the proof of Prop. 3.1.

Remark 2. Let L_{1}/K, L_{2}/K be finite separable extensions, L_{1}/K wild, L_{2}/K ferocious. Then L_{1}/K and L_{2}/K are ind-ram. Indeed, one can follow the proof of Prop. 3.1 and observe that we obtain equalities instead of inequalities.

Proposition 3.4. Let char K = 0, ζ_{p} ∈ K, L_{1} = K(x_{1}), L_{2} = K(x_{2}), x_{1}^{p} = a_{1} ∈ K, x_{2}^{p} = a_{2} ∈ K. For every i = 1, 2 suppose that either a_{i} = ρ_{i} or a_{i} = 1 + ρ_{i}, where v(ρ_{i}) ⩾ 0, ρ_{i} ∉ (K^{∗})^{p} · U_{v_{K}(ρ_{i})+1,K}. Then the following two conditions are equivalent.

1) L_{1}/K, L_{2}/K are ind-ram.

2) ∀i = 1, 2, …, p − 1: ρ_{1}/ρ_{2}^{i} ∉ (K^{∗})^{p} · U_{1,K}.

Remark. We shall give the proof only for extensions with d_{K}(L_{1}/K) = d_{K}(L_{2}/K). The proposition will be used only for extensions satisfying this condition.

Proof. Case 1: L_{1}/K is ferocious, L_{2}/K is wild. Then L_{1}/K and L_{2}/K are indram by Remark 2 above. We have p|v(ρ_{1}), p ∤ v(ρ_{2}). Therefore, p ∤ v(ρ_{1}/ρ_{2}^{i}) for any i = 1, …, p − 1, whence ρ_{1}/ρ_{2}^{i} ∉ (K^{∗})^{p} · U_{1,K}.

Case 2: L_{1}/K is wild, L_{2}/K is ferocious. Similar to the Case 1.

Case 3: both L_{1}/K and L_{2}/K are wild. In the case d_{K}(L_{1}/K) = d_{K}(L_{2}/K) = e we may assume that v(ρ_{1}) = v(ρ_{2}) = 1. Then x_{1} = τ_{1} is prime in L_{1}. In the case d_{K}(L_{1}/K) = d_{K}(L_{2}/K) = e − \frac{p−1}{p}j, j ⩾ 1, we have x_{1} = 1 + τ_{1}, v_{L_{1}}(τ_{1}) = j.

Suppose ρ_{1}/ρ_{2}^{i} ∉ (K^{∗})^{p} · U_{1,K}, i = 1, …, p − 1. Then ρ_{2} = ρ_{1} · t, t ∈ U_{K}, \overline{t} ∉ \overline{K}^{p}. This means that in L_{1} we have either a_{2} = τ_{1}^{p}t + …, or a_{2} = 1 + τ_{1}^{p}t + …, where dots denote terms of higher order. In both cases we see immediately that d_{K}(L_{1}L_{2}/L_{1}) = d_{K}(L_{2}/K) (= e or e − \frac{p−1}{p}j respectively).

Suppose ρ_{1}/ρ_{2}^{i} ∈ (K^{∗})^{p} ·U_{1,K}; clearly, this can be only when i = 1. We get either a_{2} = τ_{2}^{p}b^{p} + …, or a_{2} = 1 + τ_{2}^{p}b^{p} + … with some b ∈ U_{K}, whence d_{K}(L_{1}L_{2}/L_{1}) < d_{K}(L_{2}/K).

Case 4: both L_{1}/K and L_{2}/K are ferocious. Let first d_{K}(L_{1}/K) = d_{K}(L_{2}/K) = e. Then a_{1} = ρ_{1} ∈ U_{K}, a_{2} = ρ_{2} ∈ U_{K}, \overline{ρ_{1}}, \overline{ρ_{2}} ∉ \overline{K}^{p}. We have

\frac{ρ_{1}}{ρ_{2}^{i}} ∉ (K^{∗})^{p} · U_{1,K}, i = 1, …, p − 1 ⟺ \frac{\overline{ρ_{1}}}{\overline{ρ_{2}}^{i}} ∉ \overline{K}^{p}, i = 1, …, p − 1
⟺ \overline{L_{1}} ∩ \overline{L_{2}} = \overline{K}
⟺ \overline{ρ_{2}} ∉ \overline{L_{1}}^{p}
⟺ d_{K}(L_{1}L_{2}/L_{1}) = e = d_{K}(L_{2}/K)
⟺ L_{1}/K and L_{2}/K are ind-ram.

Next, let d_{K}(L_{1}/K) = d_{K}(L_{2}/K) < e. Then a_{1} = 1 + π^{pn}b_{1}, a_{2} = 1 + π^{pn}b_{2}, b_{1}, b_{2} ∈ U_{K}, \overline{b_{1}}, \overline{b_{2}} ∉ \overline{K}^{p}. Then

\frac{ρ_{1}}{ρ_{2}^{i}} ∉ (K^{∗})^{p} · U_{1,K}, i = 1, …, p − 1 ⟺ \frac{\overline{b_{1}}}{\overline{b_{2}}^{i}} ∉ \overline{K}^{p}, i = 1, …, p − 1,

and we proceed as above. □

Corollary 3.5. Let char K = 0, ζ_{p} ∈ K, L_{i} = K(x_{i}), x_{i}^{p} = 1 + π^{n}b_{i} ∈ K, v(b_{i}) = 0, n = \frac{p−1}{p}(e − d_{K}(L_{i}/K)), i = 1, …, m, π a prime in K.

1. Let p|n. In this case L_{1}/K, …, L_{m}/K are ind-ram if and only if d\overline{b_{1}}, …, d\overline{b_{m}} ∈ Ω_{\overline{K}/\overline{K}^{p}} are linearly independent over \overline{K}.

2. Let p ∤ n. In this case L_{1}/K, …, L_{m}/K are ind-ram if and only if d(\overline{b_{2}/b_{1}}), …, d(\overline{b_{m}/b_{1}}) ∈ Ω_{\overline{K}/\overline{K}^{p}} are linearly independent over \overline{K}.

Proof. The first assertion is proved by induction on m. The second assertion follows from the first one applied to L_{2}L_{1}/L_{1}, …, L_{m}L_{1}/L_{1}. □

The following two corollaries will not be used in the rest of the paper, and we omit their proofs.

Corollary 3.6. Suppose that L_{1}/K and L_{2}/K are ind-ram cyclic extensions of degree p, and d(L_{1}/K) = d(L_{2}/K). Let M be an intermediate field in L_{1}L_{2}/K, [M : K] = p. Then d(M/K) = d(L_{1}/K).

Corollary 3.7. Suppose that L_{1}/K and L_{2}/K are ind-ram cyclic extensions of degree p. Denote by L_{3}, …, L_{p+1} the intermediate fields in L_{1}L_{2}/K other than K, L_{1}, L_{2}, L_{1}L_{2}. Assume that d(L_{1}/K) ⩽ d(L_{2}/K). Then L_{1}/K and L_{i}/K are ind-ram for i = 3, …, p + 1. In particular, if d(L_{1}/K) = d(L_{2}/K), then L_{i}/K and L_{j}/K are ind-ram for all i ≠ j.

Partial inversion of Corollary 3.6 looks as follows.

Proposition 3.8. Let L_{1}/K, …, L_{r}/K be cyclic extensions of degree p such that [L_{1} … L_{r} : K] = p^{r}. Suppose that d(M/K) = e for any M ⊂ L_{1} … L_{r} such that [M : K] = p. Then L_{1}/K, …, L_{r}/K are ind-ram.

Proof. We apply induction on r. Let first r = 2. Withous loss of generality we may assume ζ_{p} ∈ K. Let L_{i} = K(x_{i}), x_{i}^{p} = a_{i} ∈ K, i = 1, 2. By Kummer theory, various M ⊂ L_{1}L_{2} correspond to a_{1}, a_{2}, a_{1}a_{2}, a_{1}a_{2}^{2}, …, a_{1}a_{2}^{p−1}. Since d_{K}(M/K) = e for any M, we conclude that a_{1}, a_{2}, a_{1}a_{2}, a_{1}a_{2}^{2}, …, a_{1}a_{2}^{p−1} ∉ (K^{∗})^{p} · U_{1,K}. By Proposition 3.4, L_{1}/K and L_{2}/K are ind-ram.

Let now r ⩾ 3. Suppose that L_{1}/K, …, L_{r}/K are not ind-ram. Applying the case r = 2, we obtain that d_{K}(L_{2}L_{1}/L_{1}) = ⋯ = d_{K}(L_{r}L_{1}/L_{1}) = e. Since d_{K}(L_{1}/K) = e, the extensions L_{2}L_{1}/L_{1}, …, L_{r}L_{1}/L_{1} cannot be ind-ram. By the assumption of induction, there exists some M′/L_{1} in L_{1} … L_{r}/L_{1} of degree p such that d_{K}(M′/L_{1}) < e. We have M′ = M_{1}L_{1}, [M_{1} : K] = p. Since M_{1} ⊂ L_{1} … L_{r}, d_{K}(M_{1}/K) < e, whence L_{1}/K and M_{1}/K are not ind-ram. Applying the case r = 2 once more, we come to the existence of M ⊂ L_{1}M_{1} with [M : K] = p, and d_{K}(M/K) < e. □

The following observations will be used in §4 and §7 respectively.

Lemma 3.9. Assume p − 1|e. Let L/K and K′/K be wild cyclic extensions of degree p, which are not ind-ram. Suppose that d_{K}(L/K) = e, and d_{K}(K′/K) ⩾ \frac{p−1}{p}. Then d_{K′}(LK′/K′) ⩽ e_{K′} − (p − 1). In particular, LK′/K′ is not AMR.

Proof. Direct calculation of Kummer equation yielding LK′(ζ_{p})/K′(ζ_{p}). □

Lemma 3.10. Let L/K and K′/K be ind-ram Galois extensions, L/K ferocious. Then LK′/K′ is ferocious.

Proof. The assertion can be easily reduced to the case of cyclic L/K, [L : K] = p, ζ_{p} ∈ K. In this case Lemma is obvious by Kummer theory. □

§4. General remarks on AMR extensions

In the section char K = 0; K_{n}/K is a Galois wf-extension, [K_{n} : K] = p^{n}. Denote p^{s} = e(K_{n}/K).

First we discuss the case n = 1.

Proposition 4.1. Let L/K be a cyclic extension of degree p.

1. Let L/K be wild, ζ_{p} ∈ K. In this case L/K is AMR if and only if h(L/K) ⩾ \frac{pe}{p−1} − 1.

2. Let L/K be wild, ζ_{p} ∉ K. In this case L/K is AMR if and only if h(L/K) = [\frac{pe−1}{p−1}].

3. Let L/K be ferocious. In this case L/K is AMR if and only if h(L/K) = \frac{e}{p−1}.

In general, L/K is AMR if and only if d_{K}(L/K) ⩾ e − \frac{p−1}{p}.

Proof. For a cyclic extension L/K of degree p the AMR condition is equivalent to d_{K}(L/K) − v_{K}(π_{L}) + v_{K}(π_{K}) ⩾ e, see Prop. 1.1. In the ferocious case this means that d_{L}(L/K) = d_{K}(L/K) ⩾ e, i. e., h(L/K) = e/(p − 1).

In the wild case the AMR condition is equivalent to \frac{p−1}{p}h(L/K) − \frac{1}{p} + 1 ⩾ e, i. e., h(L/K) ⩾ \frac{pe}{p−1} − 1. In the case ζ_{p} ∉ K notice that h(L/K) < \frac{pe}{p−1} and the only possible value for the ramification jump in an AMR extension is [\frac{pe−1}{p−1}]. □

Remark. The extensions of type 3 can exist only if K(ζ_{p})/K is unramified. Therefore, if e(K(ζ_{p})/K) ≠ 1, there exist no MR extensions of K.

Now we pass to the general case and start with almost obvious Lemma 4.2. Let K_{n}/K be AMR, M any intermediate field. Then M/K is AMR.

Proof. Let [M : K] = p^{m}, e(M/K) = p^{r}. Obviously, r ⩽ s. Then

d_{K}(M/K) = d_{K}(K_{n}/K) − d_{K}(K_{n}/M) ⩾ ne − 1 + \frac{1}{p^{s}} − (n − m)e = me − 1 + \frac{1}{p^{s}} .

Since p^{r}d_{K}(M/K) = d_{M}(M/K) should be an integer, we conclude that

d_{K}(M/K) ⩾ me − 1 + \frac{1}{p^{r}} ,

i. e., M/K is AMR. □

Remark. K_{n}/M need not be in general AMR.

Now we construct certain subfields K_{0} = K, K_{1}, …, K_{n−1} in K_{n}/K such that [K_{i} : K] = p^{i} by the following inductive procedure. Let G_{h} be the last non-trivial ramification subgroup in Gal(K_{n}/K); it is an elementary abelian p-group. Let H be any subgroup of order p in G_{h}; denote by K_{n−1} the corresponding intermediate field in K_{n}/K. Obviously, K_{n−1}/K is a Galois extension of degree p^{n−1}. Then this step is repeated with respect to K_{n−1}/K.

Notice that d_{K}(K_{n}/K_{n−1}) = (p − 1)\frac{h}{p^{s}}, and d_{K}(K_{n}/L) ⩽ (p − 1)\frac{h}{p^{s}}, where L is arbitrary intermediate field in K_{n}/K such that [K_{n} : L] = p. Therefore, we have the following

Lemma 4.3. 1. K_{i}/K are normal extensions of degree p^{i}, i = 1, …, n.

2. If M is an intermediate field in K_{i}/K_{i−2}, then d_{K}(K_{i}/M) ⩽ d_{K}(K_{i}/K_{i−1}), and, therefore, d_{K}(M/K_{i−2}) ⩾ d_{K}(K_{i−1}/K_{i−2}), i = 2, …, n.

Theorem 4.4. Assume p − 1|e. The extension K_{n}/K is AMR if and only if there exists a normal subextension L/K of degree p^{l}, 0 ⩽ l ⩽ n, with the following properties.

1. L/K is cyclic wild, and d_{K}(K_{i}/K_{i−1}) = e−1/p^{i−1} +1/p^{i}, i = 1, …, l, K_{i}/K denotes the subextension of L/K of degree p^{i}.

2. K_{n}/L is MR.

Proof. The existence of L with two properties immediately implies that L/K is AMR; we are to show another implication. If d_{K}(K_{i}/K_{i−1}) = e for any i, then we are done with L = K. Otherwise, consider the maximal l such that d_{K}(K_{l}/K_{l−1}) < e. Let L = K_{l}. Obviously, d_{K}(K_{n}/L) = (n − l)e, and it remains to prove the first property.

Suppose that d_{K}(K_{i}/K_{i−1}) = e for some i < l; we may assume that i is maximal possible here. Consider K_{i+1}/K_{i−1} which should be a Galois extension of degree p^{2}. If it is cyclic, then d_{K}(K_{i+1}/K_{i}) = e by Prop. 1.6, a contradiction. If it is elementary abelian, then for any intermediate field M in K_{i+1}/K_{i−1} we have d_{K}(M/K_{i−1}) = e in view of Lemma 4.3. By Prop. 3.8, K_{i}/K_{i−1} is ind-ram with any such M/K_{i−1} distinct from K_{i}/K_{i−1}, and we conclude that in this case still d_{K}(K_{i+1}/K_{i}) = e.

Thus, d_{K}(K_{i}/K_{i−1}) < e for all i ⩽ l.

Now we use induction on i to prove the following clame. For all i = 1, …, l: K_{i}/K_{i−1} is wild, d_{K}(K_{i}/K_{i−1}) = e − 1/p^{i−1} + 1/p^{i}, K_{n}/K_{i} is AMR, and K_{i}/K_{i−2} is cyclic when i ⩾ 2.

Let i = 1. If K_{1}/K were ferocious, then d_{K}(K_{1}/K) would be an integer, whence d_{K}(K_{1}/K) = e, a contradiction. Thus, K_{1}/K is wild. Next, pd_{K}(K_{1}/K) should be an integer, whence d_{K}(K_{1}/K) ⩾ e − 1 + 1/p by Lemma 4.2. Notice that p − 1 divides both e and pd_{K}(K_{1}/K) by Prop. 1.4. Since d_{K}(K_{1}/K) < e, we conclude d_{K}(K_{1}/K) = e − 1 + 1/p. We have d_{K_{1}}(K_{n}/K_{1}) ⩾ (n − 1)e_{K_{1}} − 1 + \frac{1}{p^{s−1}} , and K_{n}/K_{1} is AMR.

For i > 1 we argue similarly. Suppose that K_{i}/K_{i−2} is not cyclic. The it is elementary abelian, i. e., a composite of K_{i−1}/K_{i−2} and some cyclic M/K_{i−2}. Since d_{K}(K_{i}/K_{i−1}) > d_{K}(K_{i−1}/K_{i−2}), we should have d_{K}(M/K_{i−2}) > d_{K}(K_{i−1}/K_{i−2}), whence d_{K}(M/K_{i−2}) = e. The extensions M/K_{i−2} and K_{i−1}/K_{i−2} cannot be ind-ram, since d_{K}(K_{i}/K_{i−1}) < e. By Lemma 3.9, d_{K}(K_{i}/K_{i−1}) ⩽ e − \frac{p−1}{p^{i−1}} , a contradiction.

We have proven the claim, and the cyclicity of L/K follows by group-theoretical argument. □

Corollary 4.5. Assume p − 1|e. Let K_{n}/K be AMR. Then there exists at most one subextension M/K of degree p in K_{n}/K such that d_{K}(M/K) < e.

Proof. Suppose there exists M/K such that d_{K}(M/K) < e, and M ≠ K_{1}. Since L/K is cyclic, M ⊄ L. Then d_{K}(LM/L) < e, a contradiction with d_{K}(K_{n}/L) = (n − l)e. □

Corollary 4.6. Assume p−1|e. Suppose that K_{n}/K is a composite of two linearly disjoint extensions M/K and M′/K, and K_{n}/K is AMR. Then at least one of M/K, M′/K is MR.

Proof. Suppose that K_{n}/K is AMR. At least one of M/K, M′/K is disjoint with L/K. Assume that it is M′/K. Since LM′/L is a subextension in K_{n}/L, it is MR, and, therefore, M′/K is MR. □

§5. Cyclic AMR extensions

In this section we retain tne notation of §4 and assume additionally that K_{n}/K is cyclic.

Proposition 5.1. Let M/K be a cyclic extension of degree p^{2}, L the intermediate field.

1. Suppose that L/K is wild, h(L/K) = \frac{pe}{p−1} − i, 0 < i < \frac{min(e_{K(ζp)},p)}{e(K(ζ_{p})/K)}. Then

either M/L is wild with h(M/L) = \frac{p^{2}e}{p−1} − i, or M/L is ferocious with h(M/L) = \frac{pe}{p−1}.

2. Suppose that L/K is ferocious, h(L/K) = \frac{e}{p−1}. Then either M/L is ferocious with h(M/L) = \frac{e}{p−1}, or M/L is wild with h(M/L) = \frac{pe}{p−1}.

3. Suppose that L/K is wild, h(L/K) = \frac{pe}{p−1}. Then M/L is wild with h(M/L) =

\frac{p^{2}e}{p−1}.

Proof. Without loss of generality, we may assume that ζ_{p} ∈ K, cf. Corollary 1.3. 1. Since

d_{K}(L/K) = \frac{p − 1}{p}h(L/K) > e − \frac{p − 1}{p} min(e, p) ⩾ \frac{e}{p},

Proposition 2.2 is applicable. We have v_{K}(ρ) = i, whence v_{L}(τ) = i. Write b = π_{K}^{j}u, where u ∈ U_{K}, π_{K} = N_{L/K}π_{L}, π_{L} is a prime in L; we may assume 0 ⩽ j ⩽ p − 1. By Proposition 2.1, v_{L}(π_{K}/π_{L}^{p} − 1) ⩾ p(e − \frac{p−1}{p}i) > i. Therefore, b(1 + τ) ≡ π_{L}^{pj}u(1 + τ) mod 𝔭_{L}(i + 1). If \overline{u} ∉ (\overline{K})^{p}, then M/L is ferocious, and h(M/L) = \frac{pe}{p−1}. Otherwise, u ∈ (K^{∗})^{p} · U_{1,K} ⊂ (K^{∗})^{p} · U_{i+1,L}. Therefore, M/L is

wild, and h(M/L) = \frac{p}{p−1}e_{L} − v_{L}(τ) = \frac{p^{2}e}{p−1} − i.

2. Here the use of Proposition 2.2 yields v(1 + τ) = 0, and \overline{1 + τ} ∉ \overline{L}^{p}. Now if p|v_{K}(b), then M/L is ferocious and h(M/L) = \frac{e}{p−1}. Otherwise, M/L is wild with h(M/L) = \frac{pe}{p−1}.

3. Assuming that 1 + ρ is a prime in K, we see from Prop. 2.2 that 1 + τ is a prime in L. It follows that v_{L}(b(1 + τ)) ≡ 1 mod p, whence M/L is wild, and h(M/L) = \frac{p^{2}e}{p−1}. □

Corollary 5.2. Suppose that K_{1}/K is wild, and h(K_{1}/K) = \frac{pe}{p−1} − i, 0 < i <

\frac{min(e_{K(ζp)},p)}{e(K(ζ_{p})/K)}. Then there exist integers l and m, 1 ⩽ l ⩽ m ⩽ n, such that:

1) for 1 ⩽ j ⩽ l, K_{j}/K_{j−1} is wild, and h(K_{j}/K_{j−1}) = \frac{p^{j}e}{p−1} − i;

2) for l + 1 ⩽ j ⩽ m, K_{j}/K_{j−1} is ferocious, and h(K_{j}/K_{j−1}) = \frac{p^{l}e}{p−1};

3) for m + 1 ⩽ j ⩽ n, K_{j}/K_{j−1} is wild, and h(K_{j}/K_{j−1}) = \frac{p^{l+j−m}e}{p−1}.

Moreover, l < m unless l = m = n. In the case ζ_{p} ∉ K we have m = n.

Corollary 5.3. Suppose that K_{1}/K is MR. Then there exists an integer m, 0 ⩽ m ⩽ n such that:

1) for 1 ⩽ j ⩽ m, K_{j}/K_{j−1} is ferocious, and h(K_{j}/K_{j−1}) = \frac{e}{p−1};

2) for m + 1 ⩽ j ⩽ n, K_{j}/K_{j−1} is wild, and h(K_{j}/K_{j−1}) = \frac{p^{j−m}e}{p−1}.

In the case ζ_{p} ∉ K we have m = n.

Corollary 5.4. Suppose that K_{1}/K is AMR. Then there exist integers l and m, 1 ⩽ l ⩽ m ⩽ n such that:

1) for 1 ⩽ j ⩽ l, K_{j}/K_{j−1} is wild, and h(K_{j}/K_{j−1}) = [\frac{p^{j}e−1}{p−1}];

2) for l + 1 ⩽ j ⩽ m, K_{j}/K_{j−1} is ferocious, and h(K_{j}/K_{j−1}) = \frac{p^{l}e}{p−1};

3) for m + 1 ⩽ j ⩽ n, K_{j}/K_{j−1} is wild, and h(K_{j}/K_{j−1}) = \frac{p^{l+j−m}e}{p−1}.

Moreover, l < m unless l = m = n. In the case ζ_{p} ∉ K we have m = n.

In particular, any K_{j}/K_{j−1} is AMR, and K_{n}/K is AMR. Furthermore,

d_{K}(K_{n}/K) = ne − (1 − \frac{1}{p^{l}} )i,

where i = \frac{pe}{p−1} − [\frac{pe−1}{p−1}].

Proof. Notice that 1 < \frac{min(e_{K(ζp)},p)}{e(K(ζ_{p})/K)} unless e = e_{K} = 1. In the case e = 1 we still have

\frac{pe}{p − 1} − [\frac{pe − 1}{p − 1}] = \frac{1}{p − 1} < \frac{min(e_{K(ζ_{p})}, p)}{e(K(ζ_{p})/K)}.

Therefore, any extension K_{n}/K with AMR K_{1}/K suits the conditions of either Corollary 5.2 or Corollary 5.3. Therefore, Corollary 5.2 with i = \frac{pe}{p−1} − [\frac{pe−1}{p−1}] and Corollary 5.3 give a classification of all cyclic wf-extensions K_{n}/K with AMR K_{1}/K. (In the situation of Corollary 5.3 we have assumed l = 0.)

It remains to check that K_{n}/K is AMR. We obtain

d_{K}(K_{n}/K) = ∑_{j=1}^{n} d_{K}(K_{j}/K_{j−1}) = ∑_{j=1}^{l}(e − \frac{p − 1}{p^{j}}i) + ∑_{j=l+1}^{n} e = ne − (1 − \frac{1}{p^{l}} )i,

where i = \frac{pe}{p−1} − [\frac{pe−1}{p−1}] ⩽ 1. It follows v_{K}(D_{K_{n}/K}) = d_{K}(K_{n}/K) + 1 − \frac{1}{p^{l+n−m}} ⩾ ne = v_{K}(p^{n}). □

Theorem 5.5. The following 3 assertions are equivalent.

1. K_{n}/K is AMR.

2. K_{j}/K_{j−1} is AMR for j = 1, …, n.

3. K_{1}/K is AMR.

Proof. 1 ⟹ 2 by Theorem 4.4.

2 ⟹ 3 trivially.

3 ⟹ 1 by Corollary 5.4. □

§6. Abelian AMR extensions, p − 1|e

Lemma 6.1. Let L/K, K_{1}/K, …, K_{r}/K be ind-ram cyclic extensions of degree p. Let M/K be a cyclic extension of degree p^{2} such that L ⊂ M. Suppose that L/K is AMR, and K_{1}/K, …, K_{r}/K are MR. Next, suppose that d_{K}(M/K) ⩽ d_{K}(M′/K), where M′/K is any cyclic subsextension of degree p^{2} in MK_{1} … K_{r}/K. Then M/L, LK_{1}/L, … LK_{r}/L are ind-ram.

Proof. Denote K′ = K_{1} … K_{r}.

Case 1: M/L is not MR. Since L/K is AMR, we can apply Cor. 5.4. We obtain that d_{K}(L/K) = e − \frac{p−1}{p}i, and d_{K}(M/L) = e − \frac{p−1}{p^{2}}i, where i = \frac{pe}{p−1} − [\frac{pe−1}{p−1}]. Therefore, d_{K}(LK′/K′) = e − \frac{p−1}{p}i, and d_{K}(MK′/LK′) ⩽ e − \frac{p−1}{p^{2}}i. On the other hand, d_{K}(MK′/LK′) ⩾ e − \frac{p−1}{p^{2}}i by Prop. 1.6. We conclude that M/L is ind-ram with LK′/L. It remains to notice that LK_{1}/L, …, LK_{r}/L are obviously ind-ram and to apply Prop. 3.3.

Case 2: L/K is MR. Then d_{K}(LK′/K′) = e, whence (e. g., by Prop. 1.6) d_{K}(MK′/LK′) = e.

Case 3: L/K is not MR, whereas M/L is MR. Suppose that M/L is not ind-ram with LK′/L. By Proposition 3.8, there exists M′ ⊂ MK′ such that [M′ : L] = p, and d_{K}(M′/L) < e. Clearly, M′ ⊄ LK′, whence M′/K is cyclic of degree p^{2}. We have

d_{K}(M′/K) = d_{K}(L/K) + d_{K}(M′/L) < d_{K}(L/K) + e = d_{K}(M/K),

a contradiction. □

Theorem 6.2. Assume p − 1|e. Let E/K be an abelian p-extension. Denote by E_{1}/K the maximal subextension of E/K of exponent p. Then the following conditions are equivalent.

1. E/K is AMR.

2. E^{1}/K is AMR.

3. E = MN, where M/K and N/K are ind-ram, M/K is cyclic AMR extension, N/K is MR extension.

Proof. 1 ⟹ 3. Consider the subfield L as in Theorem 4.4. Denote by M/K the cyclic subextension of maximal degree containing L. Denote by N/K any complementary extension, i. e., any subextension of E/K such that M/K and N/K are linearly disjoint, and MN = E. Since E/L is maximally ramified, we have d_{K}(E/M) = ne, whence d_{K}(N/K) = ne, i. e., N/K is maximally ramified and ind-ram with M/K.

3 ⟹ 2. We have E_{1} = M_{1}N_{1}, where M_{1}/K, N_{1}/K are maximal subextensions of exponent p in M/K, N/K respectively. Then M_{1}/K is AMR, E_{1}/M_{1} is MR, whence E_{1}/K is AMR.

2 ⟹ 1. In view of Th. 4.4, E_{1} = K_{1}K′, where K_{1}/K is of degree p and AMR, K′/K is of exponent p and MR, K_{1}/K and K′/K are ind-ram. Let M/K be a cyclic subextension in E/K of maximal degree such that K_{1} ⊂ M, and M/N is not MR for any N, K ⊂ N ⊂ M. Let L_{1}/K be a cyclic subextension of E/K of maximal degree containing M/K. Next, choose arbitrary L_{i}/K, i = 2, …, r so that L_{i}/K are cyclic, L_{1} … L_{r} = E, [L_{1} : K] … [L_{r} : K] = [L_{1} … L_{r} : K], and L_{i} ∩ K′ = K_{i} ≠ K, i = 2, …, r.

Let L_{1}^{i}/K be the subextension of degree p^{i} in L_{1}/K. Applying Lemma 6.1, we obtain by induction on i, that L_{1}^{i+1}/L_{1}^{i}, K_{2}L_{1}^{i}/L_{1}^{i}, …, K_{r}L_{1}^{i}/L_{1}^{i} are ind-ram. Thus, K_{2}L_{1}/L_{1}, …, K_{r}L_{1}/L_{1} are MR and ind-ram.

Applying the same argument, we obtain that K_{3}L_{1}L_{2}/L_{1}L_{2}, …, K_{r}L_{1}L_{2}/L_{1}L_{2} are MR and ind-ram. Continuing by induction, we obtain that E/L_{1} is MR, whence E/K is AMR by Th. 5.5 and Th. 4.4. □

Remark. In the classical case (when the residue fields are perfect), an AMR extension should be cyclic. In the general case it is easy to construct non-cyclic AMR, and even MR, extension. In fact, let ζ_{p} ∈ K, M = k(x), N = K(y), where x^{p} ∈ a ∈ O_{K}, \overline{a} ∉ \overline{K}^{p}, y^{p} = π is a prime in K. Then M/K and N/K are ind-ram by Prop. 3.4, and MN/K is MR by Th. 6.2.

§7. Abelian AMR extensions, p − 1 ∤ e

If p − 1 ∤ e, we cannot apply Th. 4.4 any more. However, in this case the ramification of composites of ind-ram cyclic AMR extensions has quite a regular behavior.

We say that a cyclic extension L/K is purely AMR if it is AMR, and for any intermediate M the extension L/M is not MR. It is clear that any purely AMR extension is wild, and that in the case p−1 ∤ e all cyclic AMR extensions are purely AMR.

For a cyclic p-extension L/K we denote by L_{j}/K the subextension of degree p^{j} in L/K, where p^{j} ⩽ [L : K].

Lemma 7.1. Let K^{1}/K, …, K^{m}/K be ind-ram purely AMR extensions of degree p. Then K^{1} … K^{m}/K^{1} is ferocious.

Proof. The extensions K^{i}K^{1}/K^{1}, i = 2, …, m, are ind-ram and ferocious. Their composite is ferocious by Lemma 3.10. □

Proposition 7.2. Let K^{1}/K, …, K^{m}/K be purely AMR cyclic extensions. Suppose that K_{1}^{1}/K, …, K_{1}^{m}/K are ind-ram. Then:

1. K^{1}/K, …, K^{m}/K are ind-ram;

2. e(K^{1} … K^{m}/K) = max([K^{1} : K], …, [K^{m} : K]).

Proof. Let p^{r} = max([K^{1} : K], …, [K^{m} : K]); we apply induction on r.

If r = 1, we have nothing to prove, see Lemma 7.1. Let r ⩾ 2. Denote L = K_{1}^{1} … K_{1}^{m}. Without loss of generality, we may assume that for some l ⩽ m we have [K^{i} : K] ⩾ p^{2}, i = 1, …, l, and [K^{l+1} : K] = ⋯ = [K^{n} : K] = p. Now it is sufficient to prove that K^{1}L/L, …, K^{l}L/L are purely AMR, and K_{2}^{1}L/L, …, K_{2}^{l}L/L are indram. In fact, by the assumption of induction it follows that K^{1}L/L, …, K^{l}L/L are ind-ram. Notice that L/K_{1}^{i} is ferocious by Lemma 7.1 and, therefore, ind-ram with K^{i}/K_{1}^{i}, i = 1, …, l. We obtain

d_{K}(K^{1} … K^{m}/K) = d_{K}(K^{1} … K^{m}/L) + d_{K}(L/K)
= d_{K}(K^{1} … K^{l}L/L) + d_{K}(L/K)
= d_{K}(K^{1}L/L) + ⋯ + d_{K}(K^{l}L/L) + d_{K}(L/K)
= d_{K}(K^{1}/K_{1}^{1}) + ⋯ + d_{K}(K^{l}/K_{1}^{l})
+ d_{K}(K_{1}^{1}/K) + ⋯ + d_{K}(K_{1}^{m}/K)
= d_{K}(K^{1}/K) + ⋯ + d_{K}(K^{m}/K)

and

e(K^{1} … K^{m}/K) = e(K^{1} … K^{m}/L) + e(L/K) = r − 1 + 1 = r.

The fact that all K^{i}L/L are purely AMR is now clear since K^{i}/K_{1}^{i} are wild, and L/K_{1}^{i} are ferocious.

It remains to show that K_{2}^{1}L/L, …, K_{2}^{l}L/L are ind-ram. This is equivalent to the fact that K_{2}^{1}L(ζ_{p})/L(ζ_{p}), …, K_{2}^{l}L(ζ_{p})/L(ζ_{p}) are ind-ram.

For any i, the extension K_{1}^{i}(ζ_{p})/K(ζ_{p}) corresponds to Kummer equation x_{i}^{p} = 1+ π^{n}u_{i}, where u_{i} ∈ U_{K}, π is a prime in K(ζ_{p}), and n = e(K(ζ_{p})/K)(\frac{pe}{p−1} −[\frac{pe−1}{p−1}]) ⩽ p − 1 is independent from i. Applying Prop. 2.2, we see that K_{2}^{i}(ζ_{p})/K_{1}^{i}(ζ_{p}) corresponds to Kummer equation y_{i}^{p} = b_{i}(1+τ_{i}), v_{K(ζ_{p})}(τ_{i}^{p}−π^{n}u_{i}) > n, b_{i} ∈ K(ζ_{p}). Taking into account Remark after Prop 2.2, we may assume that v(b) = 0. Since K_{2}^{i}(ζ_{p})/K_{1}^{i}(ζ_{p}) is wild, we see that \overline{b_{i}} ∈ \overline{K(ζ_{p})}^{p}, i. e., b_{i} = θ_{i}^{p}(1 + πa_{i}), θ_{i}, a_{i} ∈ O_{K(ζ_{p})}. Denote τ′_{i} = (1 + τ_{i})(1 + πa_{i}) − 1 We conclude that K_{2}^{i}(ζ_{p}) = K_{1}^{i}(ζ_{p}, y′_{i}), where y′_{i}^{p} = 1 + τ′_{i}, v_{K(ζ_{p})}(τ′_{i}^{p} − π^{n}u_{i}) > n.

It follows immediately that τ′_{2}/τ′_{1}, … τ′_{m}/τ′_{1} are units in L(ζ_{p}), and (\overline{τ′_{i}/τ′_{1}})^{p} = \overline{u_{i}/u_{1}}, i = 2, …, m. By Corollary 3.5, d(\overline{u_{i}/u_{1}}) ∈ Ω_{\overline{K(ζ_{p})}/\overline{K(ζ_{p})}^{p}}, i = 2, …, m are

linearly independent over \overline{K(ζ_{p})}. There is an injective map

\overline{L(ζ_{p})} → \overline{K(ζ_{p})}
x ↦ x^{p}.

It follows that d(\overline{τ′_{i}/τ′_{1}}) ∈ Ω_{\overline{L(ζ_{p})}/\overline{L(ζ_{p})}^{p}}, i = 2, …, m are linearly independent over

\overline{L(ζ_{p})}. Applying Corollary 3.5 once more, we conclude that K_{2}^{1}L(ζ_{p})/L(ζ_{p}), …, K_{2}^{l}L(ζ_{p})/L(ζ_{p}) are ind-ram. □

In the setting of Prop. 7.2 it is easy to calculate the depth of ramification of K^{1} … K^{m}/K. Let ε = \frac{pe}{p−1} − [\frac{pe−1}{p−1}], and p^{r_{i}} = [K^{i} : K], i = 1, …, m. We obtain

d_{K}(K^{1} … K^{m}/K) = ∑_{i=1}^{m} d_{K}(K^{i}/K)

= ∑_{i=1}^{m} ∑_{j=1}^{r_{i}} d_{K}(K_{j}^{i}/K_{j−1}^{i})

= ∑_{i=1}^{m} ∑_{j=1}^{r_{i}}(e − \frac{p − 1}{p^{j}}ε)

= ∑_{i=1}^{m}(r_{i}e − (1 − \frac{1}{p^{r_{i}}} )ε).

Proposition 7.3. Let K^{1}/K, …, K^{m}/K be linearly disjoint purely AMR cyclic extensions. Then K^{1} … K^{m}/K is AMR if and only if the following 2 conditions are satisfied.

1. K_{1}^{1}/K, …, K_{1}^{m}/K are ind-ram.

2. ε ∑_{i=1}^{m}(1 − \frac{1}{p^{r_{i}}} ) ⩽ 1 − \frac{1}{p^{r}}, where p^{r_{i}} = [K^{i} : K], i = 1, …, m; r =

max(r_{1}, …, r_{m}); ε = \frac{pe}{p−1} − [\frac{pe−1}{p−1}].

Proof. If K_{1}^{1}/K, …, K_{1}^{m}/K are ind-ram, then, in view of the above calculation of d_{K}(K^{1} … K^{m}/K), the second condition is equivalent to the fact that K^{1} … K^{m}/K is AMR.

Suppose that K_{1}^{1}/K, …, K_{1}^{m}/K are not ind-ram. Take the minimal j such that K_{1}^{1}/K, …, K_{1}^{j}/K are not ind-ram. It can be shown in the spirit of Lemma 3.9 that

d_{K}(K_{1}^{1} … K_{1}^{j}/K_{1}^{1} … K_{1}^{j−1}) ⩽ e − \frac{p − 1}{p}ε − \frac{p − 1}{p}.

It follows that

d_{K}(K_{1}^{1} … K_{1}^{j}/K) ⩽ j(e − \frac{p − 1}{p}ε) − \frac{p − 1}{p} < e − 1 + \frac{1}{p^{2}} ,

i. e., K_{1}^{1} … K_{1}^{j}/K is not AMR, and K^{1} … K^{m}/K is not AMR by Lemma 4.2. □

We obtain immediately the following classification of abelian AMR extensions in the case p − 1 ∤ e.

Theorem 7.4. Assume p − 1 ∤ e. Let L/K be an abelian p-extension. Write L as K^{1} … K^{m}, where K^{i}/K are linearly disjoint cyclic, i = 1, …, m. Then L/K is AMR if and only if the following 3 conditions are satisfied.

1. The subextension of degree p in K^{i}/K is AMR for all i = 1, …, m.

2. The subextensions of degree p in K^{i}/K are ind-ram, i = 1, …, m.

3. ε ∑_{i=1}^{m}(1 − \frac{1}{p^{r_{i}}} ) ⩽ 1 − \frac{1}{p^{r}}, where p^{r_{i}} = [K^{i} : K], i = 1, …, m; r =

max(r_{1}, …, r_{m}); ε = \frac{pe}{p−1} − [\frac{pe−1}{p−1}].

§8. More on composites of AMR extensions

We return to the case e|p − 1. If we have two disjoint AMR extensions, none of which is MR, the composite cannot be AMR in view of Cor. 4.6. However, in the ind-ram case the ramification of the composite is rather regular.

In this section K_{n}/K is a cyclic extension of degree p^{n}, L_{m}/K is a cyclic extension of degree p^{m}, n ⩾ m. These extensions are always assumed to be AMR. Denote by K_{i}/K, L_{i}/K the subextensions of degree p^{i} in K_{n}/K and L_{m}/K respectively. We assume that K_{1}/K and L_{1}/K are ind-ram.

Lemma 8.1. Let n ⩾ 2. Then K_{2}L_{1}/K_{1}L_{1} is AMR.

Proof. Let L_{1}/K be ferocious. Then

d_{L_{1}}(K_{1}L_{1}/L_{1}) = d_{K}(K_{1}L_{1}/L_{1}) = d_{K}(K_{1}/K) ⩾ e_{K} − \frac{p − 1}{p} = e_{L_{1}} − \frac{p − 1}{p},

whence K_{1}L_{1}/L_{1} is AMR. By Theorem 5.5, K_{2}L_{1}/K_{1}L_{1} is AMR.

Let K_{1}/K be ferocious. Then d_{K}(K_{1}/K) = e, K_{1}L_{1}/L_{1} is MR, and, by Theorem 5.5, K_{2}L_{1}/K_{1}L_{1} is AMR.

Finally, let both L_{1}/K and K_{1}/K be wild. Then it is clear that K_{1}L_{1}/L_{1} is ferocious. We have d_{K}(K_{1}L_{1}/L_{1}) = d_{K}(K_{1}/K) ⩾ e − \frac{p−1}{p}, whence d_{L_{1}}(K_{1}L_{1}/L_{1}) ⩾ e_{L_{1}}−(p−1), and, by Prop. 1.6, d_{L_{1}}(K_{2}L_{1}/K_{1}L_{1}) ⩾ e_{L_{1}}− \frac{p−1}{p}. Since e(K_{1}L_{1}/L_{1}) = 1, we obtain that K_{2}L_{1}/K_{1}L_{1} is AMR. □

Remark. The following example shows that K_{2}/K_{1} and L_{1}K_{1}/K_{1} are in general not ind-ram. Take K = ℚ_{p}(ζ_{p^{2}})\{\{t\}\}, π = ζ_{p^{2}} − 1, K_{2} = K( \sqrt[p^{2}]{t^{p}(1 + π)}), L_{1} = K(\sqrt[p]{1 + πt}).

Lemma 8.2. In the situation of Lemma 8.1 suppose that neither of K_{1}/K, L_{1}/K, K_{2}L_{1}/K_{1}L_{1} is MR. Then K_{1}(ζ_{p}) = K(ζ_{p}, \sqrt[p]{1 + ρ}), K_{2}L_{1}(ζ_{p}) = K_{1}L_{1}(ζ_{p}, \sqrt[p]{1 + τ}), v(τ^{p} − ρ) > v(ρ) > 0.

Proof. Direct calculation using Prop. 2.2. □

Lemma 8.3. Let m ⩾ 2. Then K_{2}L_{2}/K_{1}L_{1} is a composite of two cyclic ind-ram AMR extensions of degree p.

Proof. 1. Suppose first that at least one of K_{1}/K, L_{1}/K is MR. Then our assertion follows immediately from Theorem 6.2.

2. Now both K_{1}/K and L_{1}/K are not MR. Consider all subextensions of degree p in K_{2}L_{2}/K_{1}L_{1}: M_{1}/K_{1}L_{1}, …, M_{p+1}/K_{1}L_{1}. Suppose that all they are MR. Then any two them are ind-ram in view of Prop. 3.8.

3. Suppose that some of M_{1}/K_{1}L_{1}, …, M_{p+1}/K_{1}L_{1} are MR, and some not; say, M_{1}/K_{1}L_{1} is MR, and M_{2}/K_{1}L_{1} is not MR. Changing K_{2} and L_{2} inside K_{2}L_{2}, we may assume without loss of generality that M_{1} = K_{2}L_{1}, M_{2} = K_{1}L_{2}. It follows that K_{2}/K_{1} is MR. Applying results of §4, we conclude that K_{2}(ζ_{p}) = K_{1}(ζ_{p}, \sqrt[p]{a}), where a ∈ O_{K_{1}(ζ_{p})}, \overline{a} ∉ (\overline{K_{1}(ζ_{p})})^{p}. It follows that K_{2}L_{1}/K_{1}L_{1} is ferocious. On the other hand, K_{1}L_{2}/K_{1}L_{1} is AMR (by Lemma 8.1) but not MR. Therefore, it is wild and ind-ram with K_{2}L_{1}/K_{1}L_{1}.

4. It remains to consider the case when all M_{1}/K_{1}L_{1}, …, M_{p+1}/K_{1}L_{1} are not MR.

Let K′ = K(ζ_{p}). We have K_{1}K′ = K′(\sqrt[p]{1 + ρ_{1}}), L_{1}K′ = K′(\sqrt[p]{1 + ρ_{2}}), where

v_{K′}(ρ_{1}) = v_{K′}(ρ_{2}) = i = (\frac{pe}{p − 1} − [\frac{pe − 1}{p − 1}])e(K′/K).

Since K_{1}/K and L_{1}/K are ind-ram, by Prop. 3.4, \overline{ρ_{1}/ρ_{2}} ∉ (\overline{K′})^{p}. By Lemma 8.2, K_{2}L_{1}K′ = K_{1}L_{1}K′(\sqrt[p]{1 + τ_{1}}), K_{1}L_{2}K′ = K_{1}L_{1}K′(\sqrt[p]{1 + τ_{2}}), v_{K′}(τ_{1}^{p} − ρ_{1}) > i, v_{K′}(τ_{2}^{p} − ρ_{2}) > i. Therefore,

\overline{(\frac{τ_{1}}{τ_{2}} )} = \sqrt[p]{\frac{ρ_{1}}{ρ_{2}}} ∉ (\overline{K_{1}L_{1}K′})^{p}.

Indeed, (\overline{K_{1}L_{1}K′})^{p} ⊂ \overline{K′}. By Prop. 3.4, we conclude that K_{2}L_{1}/K_{1}L_{1} and K_{1}L_{2}/K_{1}L_{1} are ind-ram. □

Theorem 8.4. Let n > m. Then K_{n}L_{m}/K_{n−1}L_{m} is AMR.

Proof. Application of Lemma 8.3 decreases both m and n by 1. In the case m = 1 Theorem follows from Lemma 8.1. □

Theorem 8.5. Let n = m. Then K_{n}L_{m}/K_{n−1}L_{m−1} is a composite of two ind-ram AMR cyclic extensions of degree p.

Proof. Apply n times Lemma 8.3. □

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Permanent adress: Department of Mathematics and Mechanics, St. Petersburg University, Bibliotechnaya pl. 2, 198904 St. Petersburg, Russia E-mail: sergei@vostokov.usr.pu.ru, igor@zhukov.pdmi.ras.ru