ELIMINATION OF WILD RAMIFICATION

M. V. Koroteev, I. B. Zhukov

September 20, 1999

To appear in Algebra i Analiz/St. Petersburg Math. J. 11, No.6

Introduction

In 1972 Epp [E] proved a theorem asserting that in an arbitrary finite extension of discretely valued fields L/K one can eliminate wild ramification, i. e. to guarantee validity of e(k′L/k′K) = 1 for some finite extension k′/k, where k is a constant subfield: maximal subfield of K with perfect residue field (k is canonical in the mixed characteristic case). In the papers by Ponomaryov [P1,P2] some inconsistency of the proof in [E] was removed (which took place when K is of prime characteritic). Also Ponomaryov proved a number of strong generalizations of the elimination theorem for arbitrary extensions of discretely valued fields of prime characteristic.

In [Z1] a variant of the proof of Epp’s theorem in the mixed characteristic case was given. It has been also shown how one deduces from it a certain advanced version of Parshin’s classification theorem for multidimensional local fields [Pa].

In the present work we consider only the case of a finite extension of complete discretely valued fields. In the case of prime characteristic we choose a constant subfield k in a given complete discretely valued field K, and we assume (outside §3) that k is fixed. We prove the following variant of Epp’s theorem.

Theorem 1. Let L/K be any finite extension of complete discretely valued fields, k the constant subfield in K. Then there exists a finite solvable extension k′/k such that k′L/k′K is weakly unramified, i. e. e(k′L/k′K) = 1.

Let e_{L} = p^{h} · e_{t} where (p, e_{t}) = 1. Then there exists desired k′/k that is a composite of a cyclic extension and an extension of degree not exceeding (p − 1)^{h+1}[L : K]e_{K}.

The proof proceeds for the equal and mixed characteristic cases in parallel. It is based on the splitting of an arbitarary finite extension into a tower of an “almost constant” and “infernal” extensions. (An extension is said to be almost constant if it can be embedded into a composite of a constant extension and absolutely unramified one. An extension is said to be infernal if it possesses no non-trivial almost constant subextension.) We remark that elimination of wild ramification

Typeset by AMS-TEX

Partially supported by a grant of Russian ministry of general and professional education No.

97-01.1-1 and a grant of RFBR (project No. 97-01-00058-a).

in an infernal extension yields a ferociously ramified extension of the same degree, that means an extension of ramification index 1 and purely inseparable extension of the residue fields.

The key assertion in the proof is the following theorem which is also of independent interest for the ramification theory.

Theorem 2. Let L/Kbe an infernal extension, k the constant subfield of K, C/k an arbitrary deeply ramified^{1} extension. Then there exists a finite subextension k′/k in C/k such that k′L/k′K is ferociously ramified.

Thus, in particular, k′/k can be chosen, e. g., inside any given ramified extension with the Galois group isomorphic to ℤ_{p}.

The first 2 sections of this paper contain preliminary results. §3 is devoted to the proof of Theorems 1 and 2. In the last subsection of §3 we deduce also Theorems 3 and 4 which are new variants of structure theorems for complete fields from [Z1].

Namely, a field K is said to be standard if e(K/k) = 1 where k is the constant subfield. Notice that a multidimensional local field K of characteristic 0 with the first residue field of characteristic p and with perfect last residue field is standard if and only if it takes the shape l{{t_{1}}} … {{t_{n}}}.

Theorem 3. Let e(K/k) = p^{h}e_{t} where (p, e_{t}) = 1. Then there exists a finite solvable constant extension L/K such that L is a standard field. Moreover, one may assume L = L_{1}L_{2}, where L_{1}/K is a cyclic p-extension, whereas [L_{2} : K] divides (p − 1)^{h+1}e(K/k). Furthermore, one may assume L_{1} = lK where l/k is a finite subextension of a given ramified constant ℤ_{p}-extension C/k.

We recall that in the mixed characteristic case a subfield k is determined canonically; in particular, for a multidimensional local field K with the finite last residue field, the subfield k coincides with the algebraic closure of ℚ_{p} in K.

A field is said to be almost standard, if some finite unramified extension of it is a standard field.

Theorem 4. Let k be the constant subfield of K, C/k is a deeply ramified extension. Then there exists a finite subextension l/k in C/k such that lK is almost standard.

Corollary. Let K be a multidimensional loacal field of characteristic 0 with the first residue field of characteristic p > 0, k its constant subfield. Suppose that C/k is a deeply ramified extension. Then there exists a finite subextension l/k in C/k such that lK is almost standard. In particular, there exists a semi-ramified extension L/K such that L(ζ_{p^{n}}) is a standard field for some positive integer n.

In §4 we show that in the equal characteristic case by means of a suitable choice of the constant subfield one can force any given Galois p-extension to be infernal. This permits one to eliminate ramification by means of a cyclic k′/k (under some obvious restrictions). In particular, the following theorem takes place.

Theorem 5. Let char K = p and K contain all roots of unity of degree prime to p. Then for any finite separable extension L/K there exists a constant subfield k in K and a cyclic extension k′/k such that e(k′L/k′K) = 1.

1Deeply ramified extensions were introduced in [CG] and thoroughly studied in [F]. In the present paper the notion of a deeply ramified extension is slightly more general, see §1.

The second named author expresses his gratitude to Max-Planck-Institut für Mathematik where the final version of this work was done. The authors are grateful to Prof. F. Lorenz for Lemma 3.3.3, which led to considerable improvement of results in this paper.

Notation

• p always denotes a certain fixed prime number;

• ζ_{m}: a primitive root of unity of degree m in the separable closure of the given field;

• ℘: F → F: the map x ↦ x^{p} − x in the given field F of characteristic p; For a complete discretely valued field K with the residue field of characteristic

p:

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

• O_{K}: the valuation ring in K;

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

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

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

• R_{K}: the subgroup of Teichmüller representatives, i. e. the subgroup of K^{∗} which consists of representatives of all non-zero elements of the maximal perfect subfield in \overline{K}.

• A solvable extension means a subextension of a finite Galois extension with solvable Galois group.

§1. Basic definitions and constructions

1.1. Almost constant extensions. Let K be a complete discretely valued field of any characteristic with the residue field \overline{K} of characteristic p > 0.

If char K = p, we fix a base subfield B in K, which is supposed to be complete with respect to v_{K} and to have the residue field 𝔽_{p}. (The possible base subfields are exactly the subfields 𝔽_{p}((τ)) where v_{K}(τ) > 0.) If char K = 0, we assume B = ℚ_{p}. Denote by v: K^{∗} → ℚ the non-normalized extension of v_{B} onto K; denote also e_{K} = e_{K/B} = 1/v(π) where π is a prime in K. In the case char K = 0 we have, as usual, e_{K} = v_{K}(p).

Let F = ⋂ \overline{K}^{p^{i}} be the maximal perfect subfield of \overline{K}. If char K = 0, denote by k the set of all x ∈ K which are algebraic over the fraction field k_{0} of W (F). In the equal characteristic case the definition of k is the same, if we take for k_{0} the completion of B(R_{K}). Obviously, in both cases k is a complete subfield of K with perfect residue field, and it is maximal with respect to this property. The subfield k will be referred to as the constant subfield of K.

Example. Consider a standard n-dimensional local field K = k{{t_{1}}} … k{{t_{n−1}}} where k is a complete discretely valued field of characteristic 0 with a perfect residue field of characteristic p. Then the constant subfield of K coincides with k.

A finite extension L/K is said to be constant, if L = lK where l is an algebraic extension of k. Clearly, in this case l is the constant subfield of L. Notice that separable constant extensions of K are exactly the extensions of the kind K(a)/K, where a is algebraic and separable over k. A finite extension will be referred to as almost constant if it lies in a composite of a constant extension and an absolutely unfamified one. Equivalently, L/K is almost constant, if K′L/K′ is constant for some unramified K′/K. Notice the following properties.

1. A composite of two almost constant extensions is almost constant. Therefore, it makes sense to consider the maximal almost constant subextension in a given finite extension.

2. Any intermediate extension in an almost constant extension L/K is almost constant as well. In fact, let K ⊂ K′ ⊂ L′ ⊂ L. We have L ⊂ CU where C/K is a constant extension, and U/K an unramified one. Then L′ ⊂ (CK′)(UK′), CK′/K′ is a constant extension, UK′/K′ an unramified one.

3. Any weakly ramified extension is almost constant.

4. Let char K = p, L/K a cyclic extension of degree p. Then L/K is almost constant, if and only if L = K(x), x^{p} − x ∈ k + O_{K}.

5. Let char K = 0, ζ_{p} ∈ K, L/K a cyclic extension of degree p. Then L/K is almost constant, if and only if L = K(x), x^{p} ∈ k^{∗} · (1 + 𝔐_{K}^{a}), a = \frac{p}{p−1}e_{K}.

We shall say that K is standard, if e_{K/k} = 1. It is obvious that any unramified or constant or ferociously ramified extension of a standard field is standard.

Further, K is said to be almost standard, if there exists an unramified extension L/K such that L is standard. It is easy to prove that an almost constant or ferociously ramified extension of an almost standard field is almost standard.

1.2. Infernal extensions. We assign the name infernal to the opposite type of extensions. Namely, a finite extension L/K will be referred to as infernal if the only almost constant subextension in L/K is K itself.

1. If L/K is infernal, then any intermediate extension in L/K is infernal.

Suppose, there exist K′ and L′ such that K ⊂ K′ ⊂ L′ ⊂ L, and L′/K′ is almost constant. Then L′ ⊂ CU, C/K′ is constant, U/K′ is unramified. Obviously, C = C_{0}K′ where C_{0}/K is a constant extension. Next, U = U_{0}K′ where U_{0}/K is an unramified extension. The extension C_{0}U_{0}/K is linearly disjoint with K′/K since K′/K is infernal. We conclude that L′ = L_{0}K′ where L_{0} ⊂ C_{0}U_{0}. Thus, L_{0}/K is an almost constant subextension in L/K, whence L_{0} = K, and L′ = K′.

2. If L/K is infernal, then K′L/K′ is also infernal for any almost constant extension K′/K.

In fact, let E/K′ be an almost constant subextension in K′L/K′. Then (E ∩ L)/K is almost constant and infernal at the same time, whence E ∩ L = K. This yields [LK′ : E] = [L : K] ⩾ [LK′ : K′], and E = K′.

1.3. Jumps and depth of ramification. Let L/K be a finite Galois extension, σ ∈ Gal(L/K) a non-trivial automorphism. Then the (absolute) ramification number s(σ) ∈ ℚ is defined as min{ v(σ(a)a^{−1} − 1) | a ∈ O_{L} }. All elements of { s(σ) | σ ∈ Gal(L/K), σ ≠ 1 } are said to be the ramification jumps of L/K. If L/K is a normal inseparable extension, the ramification jumps of L/K are defined as {jumps of L_{0}/K} ∪ {∞}, where L_{0}/K is the maximal separable subextension. If L/K possesses an only ramification jump (this is always the case if [L : K] is prime), we denote this jump by h(L/K).

Let L/K be a finite separable extension. Then the depth of ramification is defined:

d(L/K) = inf{ v(Tr_{L/K}(y)/y) | y ∈ L^{∗} }.

It should be noted that for any intermediate N:

d(L/K) = d(L/N) + d(N/K)

[H, 2–4], and for a weakly ramified L/K we have d(L/K) = 0 [H, 2–12]. Finally, for a cyclic L/K of degree p we have d(L/K) = (p − 1)h(L/K), see [H, 2–10].

Lemma 1.3.1. Let L_{1}/K and L_{2}/K be cyclic extensions of degree p. Then

h(L_{1}L_{2}/L_{1}) = h(L_{2}/K) − h(L_{1}/K) + h(L_{1}L_{2}/L_{2}).

Proof. We have

h(L_{1}L_{2}/L_{1}) + h(L_{1}/K) = \frac{1}{p − 1}(d(L_{1}L_{2}/L_{1}) + d(L_{1}/K))

= \frac{1}{p − 1}(d(L_{1}L_{2}/K))

= \frac{1}{p − 1}(d(L_{1}L_{2}/L_{2}) + d(L_{2}/K))

= h(L_{1}L_{2}/L_{2}) + h(L_{2}/K). □

Definition. An (infinite) extension l/k is said to be deeply ramified if the inertia degree [\overline{l} : \overline{k}] is finite and at least one of the following two conditions is satisfied.

1) l/k possesses finite separable extensions of arbitrarily big depth of ramification, i. e. the set { d(k′/k) | k ⊂ k′ ⊂ l, k′/k separable } is not bounded.

2) [l : k]_{insep} = ∞.

It is easy to see that in the case of separable L/K we have the usual notion of deeply ramified extension, compare [F].

Lemma 1.3.2. Let l/k be deeply ramified, k′/k a finite subextension in l/k. Then l/k′ is deeply ramified.

1.4. On the ramification in an extension of degree p. Let L/K be a cyclic extension of degree p. In the case char K = p we have L = K(γ), where ℘(γ) = a ∈ K. If v_{K}(a) > 0, then by Hensel lemma the polynomial ℘(x) − a splits completely over K, and the extension L/K is therefore trivial. This means also that 𝔐_{K} ⊂ ℘(K). Obviously, in the right hand side of Artin–Schreier equation ℘(γ) = a the element a can be replaced by any element of the class a + ℘(K).

In the case char K = 0 suppose that ζ_{p} ∈ K. Then any cyclic extension L/K of degree p corresponds to an equation x^{p} = a ∈ K. Here a can be replaced by any element from aK^{∗p}.

Proposition 1.4. Let char K = p, L = K(γ) a non-trivial extension, where ℘(γ) = a ∈ K. In this case:

1) if v_{K}(a) = 0, then L/K is unramified;

2) if v_{K}(a) < 0 and (p, v_{K}(a)) = 1, then L/K is totally ramified and h(L/K) = −v(a)/p;

3) if v_{K}(a) < 0, p | v_{K}(a) and v(a) = max{v(b) | b ∈ a + ℘(K)}, then L/K is ferociously ramified and h(L/K) = −v(a)/p.

Let char K = 0, L = K(γ) a non-trivial extension, γ^{p} = a ∈ K. In this case:

1) if v(a − 1) = \frac{p}{p−1}, then L/K is unramified;

2) if 0 < v(a − 1) < \frac{p}{p−1} and (p, v_{K}(a − 1)) = 1, then L/K is totally ramified and h(L/K) = \frac{1}{p−1} − \frac{1}{p}v(a − 1);

3) if 0 < v(a−1) < \frac{p}{p−1}, p | v_{K}(a−1), and v(a−1) = max\{v(b−1) | b ∈ a(K^{∗})^{p}\}, then L/K is ferociously ramified and h(L/K) = \frac{1}{p−1} − \frac{1}{p}v(a − 1);

4) if v_{K}(a) = 0, a ∉ K^{p}, then L/K is ferociously ramified and h(L/K) = \frac{1}{p−1};

5) if v_{K}(a) > 0, p | v_{K}(a), then L/K is totally ramified and h(L/K) = \frac{1}{p−1}.

Proof. Let e = e(L/K), f = f(L/K). Since ef = p, and p is prime, we have either e = 1, f = p (the extension is weakly unramified), or e = p, f = 1 (the extension is totally ramified). Denote by π any prime element of K.

Consider first the case char K = p. For subcases 1), 2) see [FV, Prop. III.2.5]. (In the first subcase notice also that \overline{a} ∉ ℘(\overline{K}) in view of Hensel lemma.) Suppose now that v_{K}(a) < 0, p | v_{K}(a) and v(a) = max{v(b) | b ∈ a+℘(K)}. Let a = cπ^{−np}, where v_{K}(c) = 0. If c̄ = (b̄)^{p}, b ∈ F, then v_{K}(a − (bπ^{−n})^{p} + bπ^{−n}) > v_{K}(a), a contradiction with the choice of a. On the other hand, denote b = π^{n}γ ∈ O_{L}. Then \overline{b}^{p} = \overline{c}, whence \overline{L}/\overline{K} is purely inseparable, and \overline{L} = \overline{K}(\overline{b}).

Now consider the case char K = 0. We treat consequently all 5 subcases.

1) Let γ = 1+π^{\frac{e_{k}}{p−1}} x, a = 1+π^{\frac{pe_{k}}{p−1}} b. We obtain v(x) = 0, v(b) = 0, x^{p}+ \frac{p}{π^{e}} x = b. The latter equation should not be solvable in K, whence \overline{x}^{p} + \frac{p}{π^{e}} \overline{x} = \overline{b} has no roots in \overline{K}. This equation is separable, and L/K is therefore unramified.

2) Since v(γ − 1) = v(a − 1)/p, L/K is totaly ramified. Let σ be any generator of Gal(L/K). Then γ^{σ−1} is a primitive pth root of unity, whence

v((γ − 1)^{σ−1} − 1) = \frac{1}{p − 1} − \frac{1}{p}v(a − 1).

Since (v_{L}(γ − 1), p) = 1 one can find integral m and n such that π_{L} = (γ − 1)^{m}/π^{n} is a prime in L. Then

v(π_{L}^{σ−1} − 1) = \frac{1}{p − 1} − \frac{1}{p}v(a − 1),

and we are done since O_{L} = O_{K}[π_{L}].

3) Introduce d ∈ O_{K} and t ∈ O_{L} via a = 1 + π^{pl}d, γ = 1 + π^{l}t, where l = v_{K}(a − 1)/p. By the maximality of v(a − 1), we have \overline{d} ∉ \overline{K}^{p}. On the other hand, t^{p} ≡ d mod 𝔐_{L}, whence L/K is ferociously ramified, and O_{L} = O_{K}[t]. We obtain (in the above notation):

h(L/K) = v(t^{σ−1} − 1)) = v((γ − 1)^{σ−1} − 1) = \frac{1}{p − 1} − \frac{1}{p}v(a − 1).

4), 5) Similarly to 3) and 2) respectively. □

§2. Infernal jumps

Throughout this section K denotes a standard complete discretely valued field with the residue field of characteristic p, B a fixed base subfield in K, k the corresponding constant subfield in K, v = v_{B}.

2.1. Filtration on the additive group of K in the case char K = p.

Definition. Let char K = p. For any α ∈ ℚ, α ⩾ 0, define Ξ_{α} as the subgroup in K which is generated by all a^{p^{i}}c where i ⩾ 0, a ∈ O_{K}, c ∈ k, v(c) ⩾ −p^{i+1}α. We have obviously Ξ_{0} = O_{K} + k. Next, Ξ_{α} ⫋ Ξ_{β} for α < β. For a ∈ K, put w_{K}(a) = inf{ α | a ∈ Ξ_{α} } = min{ α | a ∈ K_{α} }.

Remark. The filtration (Ξ_{α} + ℘(K))_{α} coincides with the infernal part of the filtration on the additive group of K which is dual (in the sense of Artin-Schreier duality) to the ramification filtration on the Galois group of the maximal abelian extension of K of exponent p introduced in [Z2].

Lemma 2.1.1. Let a_{1}, a_{2} ∈ O_{K}, c_{1}, c_{2} ∈ k, v_{k}(c_{1}) = v_{k}(c_{2}) = i. For some integral k_{1} ⩾ k_{2} ⩾ 0 denote

s = a_{1}^{p^{k_{1}}} c_{1} + a_{2}^{p^{k_{2}}} c_{2}.

Then there exist a′_{1}, a′_{2} ∈ U_{K} ∪ {0}, c′_{1}, c′_{2} ∈ k, v_{k}(c′_{1}) ⩾ i, v_{k}(c′_{2}) > i such that

s = a′_{1}^{p^{k_{2}}} c′_{1} + a′_{2}^{p^{k_{2}}} c′_{2}.

Proof. We may assume that c_{2} ≠ 0. The residue class of c_{1}/c_{2} belongs to the maximal perfect subfield in \overline{K}, hence there exists r ∈ R_{K} such that r̄^{p^{k_{2}}} = c̄_{1}/c̄_{2}. Now take a′_{1} = a_{1}^{p^{k_{1}−k_{2}}} + a_{2}/r, a′_{2} = a_{2}, c′_{1} = c_{1} and c′_{2} = c_{2} − c_{1}/(r^{p^{k_{2}}} ). If we have 0 < v_{K}(a′_{1}) < ∞, then instead of a′_{1} and c′_{1} one must take a′_{1}π_{k}^{−v_{K}(a′_{1})} and c′_{1}π_{k}^{v_{K}(a′_{1})p^{k_{2}}} respectively, where π_{k} is a prime element in k. If necessary, the same can be done for the pair a′_{2}, c′_{2}. Now the condition v_{k}(c′_{2}) > i is satisfied by the choice of r, and the equality in Lemma can be easily checked. □

Corollary 2.1.2. Let c_{j} be certain fixed elements of k with v_{k}(c_{j}) = j for all integral j.

1. For any α > 0 we have

Ξ_{α} = { ∑_{j≫−∞} a_{j}^{p^{i(j)}}c_{j} | a_{j} ∈ O_{K} } = { ∑_{j≫−∞} a_{j}^{p^{i(j)}}c_{j} | a_{j} ∈ U_{K} ∪ {0} },

where i(j) = min{ i ⩾ 0 | j/e_{k} ⩾ −p^{i+1}α }.

2. For any a ∈ K the value w_{K}(a) is equal to the minimal w such that there exists an expansion

(1)  a = ∑_{{v(a)⩽j<0 \atop j∈e_{k}^{−1}ℤ}}(a_{j})^{p^{⌈log_{p}( \frac{−j}{w} )−1⌉}}c_{j} + a_{0},

where a_{j} ∈ U_{K} ∪ {0} for v(a) ⩽ j < 0, a_{0} ∈ O_{K}. If there is no minimal w, then w_{K}(a) = 0. Here ⌈x⌉ denotes the minimal non-negative integer which is ⩾ x.

Lemma 2.1.3. Let a ∈ U_{K}, \overline{a} ∉ \overline{K}^{p}, c ∈ k, v_{k}(c) = j_{0}, u ∈ K, v_{k}(u − a^{p^{i}}c) > j_{0}. Then w_{K}(u) ⩾ \frac{−j_{0}}{p^{i+1}e_{k}}.

Proof. Let w_{K}(u) < \frac{−j_{0}}{p^{i+1}e_{k}} which is equivalent to u ∈ Ξ_{α} for some

(2)  α < \frac{−j_{0}}{p^{i+1}e_{k}} .

Corollary 2.1.2 implies

u = ∑_{j=j_{0}}^{∞} a_{j}^{p^{i(j)}}c_{j},

where c_{j_{0}} = c, and c_{j} with j > j_{0} are arbitarily chosen. After division by c we obtain

a^{p^{i}} ≡ a_{j_{0}}^{p^{i(j_{0})}} mod 𝔐_{K},

and (\overline{a})^{p^{i}} ∈ \overline{K}^{p^{i(j_{0})}}. In view of \overline{a} ∉ \overline{K}^{p}, we obtain i(j_{0}) ⩽ i, i. e. j_{0}/e_{k} ⩾ −p^{i+1}α, contradicting (2). □

Lemma 2.1.4. Consider b_{0} = a^{p^{n}}c where 0 ⩽ n ∈ ℤ, a ∈ U_{K}, \overline{a} ∉ \overline{K}^{p}, c ∈ k, v(c) < 0. Then

w_{K}(b_{0}) = \frac{−v(c)}{p^{n+1}}.

Next, for any b ∈ b_{0} + ℘(K) we have

w_{K}(b) ⩾ \frac{−v(c)}{p^{n+1}}.

Proof. The first assertion follows immediately from Lemma 2.1.3. To prove the second assertion, we assume first that either n = 0 or p ∤ v_{k}(c). Suppose there exists b ∈ b_{0} + ℘(K) with w_{K}(b) < w_{K}(b_{0}). Among all such b we choose an element with the maximal v(b); we have b = b_{0} + d^{p} − d, d ∈ K. It can be seen from the Lemma 2.1.3 that v(d) > v(b_{0})/p implies w_{K}(b_{0} + d^{p} − d) ⩾ \frac{−v(c)}{p^{n+1}}. Furthermore, the case v(d) = v(b_{0})/p is possible only if p | v_{k}(c), whence n = 0. In this case we have

\overline{(\frac{b_{0} + d^{p} − d}{c})} = \overline{a} + \overline{(\frac{d^{p}}{c} )} ∉ \overline{K}^{p},

and this is also incompatible with the hypothesis w_{K}(b) < w_{K}(b_{0}). Finally, let v(d) < v(b_{0})/p. We can write d as a_{0}^{p^{n_{0}}} c_{0} + d′ where a_{0} ∈ U_{K}, ā_{0} ∉ \overline{K}^{p}, c_{0} ∈ k and v(d′) > v(d) = v(c_{0}). Then b = b_{0} + d^{p} − d = a_{0}^{p^{n_{0}+1}}c_{0}^{p} + …, where dots denote terms of higher order. From Lemma 2.1.3 we have

(3)  w_{K}(b) ⩾ w_{K}(a_{0}^{p^{n_{0}+1}}c_{0}^{p}) = w_{K}(a_{0}^{p^{n_{0}}} c_{0}).

Put b_{1} = b − ℘(a_{0}^{p^{n_{0}}} c_{0}). Then it is seen from (3) that w_{K}(b_{1}) ⩽ w_{K}(b) whereas v(b_{1}) > v(b). This contradicts to the choice of b.

Now consider the remaining case when n > 0 and p | v_{k}(c). Use induction on n. Since p | v_{k}(c), we have c = c_{1}^{p} + c′, where c_{1}, c′ ∈ k, and v(c′) > v(c). Then

b_{0} = a^{p^{n}}c_{1}^{p} + a^{p^{n}}c′ = a^{p^{n−1}}c_{1} + ℘(a^{p^{n−1}}c_{1}) + a^{p^{n}}c′,

whence b = b_{1} + a^{p^{n}}c′, b_{1} ∈ a^{p^{n−1}}c_{1} + ℘(K). We have

w_{K}(b_{1}) ⩾ \frac{−v(c_{1})}{p^{n}} = \frac{−v(c)}{p^{n+1}}

by the assumption of induction, and

w_{K}(a^{p^{n}}c′) = \frac{−v(c′)}{p^{n+1}} < \frac{−v(c)}{p^{n+1}}

by the first assertion of Lemma. This proves the second assertion. □

Lemma 2.1.5. Let B be a class in K/℘(K). Then there exists b ∈ B such that w_{K}(b) ⩽ w_{K}(b′), and v(b) ⩾ v(b′) for any b′ ∈ B.

Proof. Suppose first that B contains some a ∈ O_{K}. Then the assertion is trivial (either a or 0 suits for b). Otherwise consider the subset B_{0} in B which consists of all elements b ∈ B with the maximal v(b). It is clear from the definition that w_{K} takes finitely many values on B_{0}; fix an element b ∈ B_{0} with the minimal w_{K}(b).

Suppose that w_{K}(b) > w_{K}(b′) for some b′ ∈ B. Choose such a b′ with the maximal v(b′). Then b′ = a^{p^{n}}π^{pr} + d, where π is a prime in k, r = v_{k}(b′), a ∈ U_{K}, n a positive integer, v(d) > v(b′), either \overline{a} ∉ \overline{K}^{p} or a = 1. Consider

b′′ = b′ + ℘(−a^{p^{n−1}}π^{r}).

It follows from Lemma 2.1.3 that

w_{K}(b′) ⩾ w_{K}(a^{p^{n}}π^{pr}) = w_{K}(a^{p^{n−1}}π^{r}).

Therefore, w_{K}(b′′) ⩽ w_{K}(b′) < w_{K}(b), whereas v(b′′) > v(b′). This contradicts to the choice of b′. □

2.2. Filtration on K^{∗} in the case char K = 0.

In this subsection char K = 0. Denote ˜℘_{K} = { a ∈ K | v(a) ⩾ p/(p − 1) }. For a non-negative integer m and a rational number α such that 0 ⩽ α ⩽ \frac{1}{p−1} define Ξ(m, α) as the subgroup in K which is generated by ˜℘_{K} and all elements of the kind a^{p^{i}}c where i ⩾ 0, a ∈ U_{K}, c ∈ k, v(c) ⩾ max(m/e_{k}, \frac{p}{p−1} − p^{i+1}α).

Lemma 2.2.1. Let a_{1}, a_{2} ∈ U_{K}, c_{1}, c_{2} ∈ k, v(c_{1}) = v(c_{2}) = m/e_{k} ⩾ \frac{p}{p−1} − p^{i+1}α. For an integer i ⩾ 0 denote

s = a_{1}^{p^{i}}c_{1} + a_{2}^{p^{i}}c_{2}.

Fix c ∈ k, v(c) = m/e_{k}. Then there exists a ∈ U_{K} ∪ {0} such that

s ≡ a^{p^{i}}c mod Ξ(m + 1, α).

Proof. As in the proof of Lemma 2.1.1, we take r_{1}, r_{2} ∈ R_{K} such that r̄_{1}^{p^{j}} = c̄_{1}/c̄, r̄_{2}^{p^{j}} = c̄_{2}/c̄.

Denote a = a_{1}r_{1} + a_{2}r_{2}. Then

a^{p^{i}}c ≡ a_{1}^{p^{i}}cr_{1}^{p^{i}} + a_{2}^{p^{i}}cr_{2}^{p^{i}} + ∑_{j=1}^{p−1} \frac{1}{p}\binom{p}{j}(a_{1}r_{1})^{jp^{i−1}}(a_{2}r_{2})^{(p−j)p^{i−1}}pc mod p^{2}.

We have cr_{ν}^{p^{i}} ≡ c_{ν} mod 𝔐_{k}, ν = 1, 2, whence

s ≡ a_{1}^{p^{i}}cr_{1}^{p^{i}} + a_{2}^{p^{i}}cr_{2}^{p^{i}} mod Ξ(m + 1, α).

Next, for any j we have (a_{1}r_{1})^{jp^{i−1}}(a_{2}r_{2})^{(p−j)p^{i−1}}pc ∈ Ξ(m + 1, α), since v(pc) = 1 + m/e_{k} ⩾ (m + 1)/e_{k}, and v(pc) = 1 + m/e_{k} ⩾ 1 + \frac{p}{p−1} − p^{i+1}α together with v(pc) ⩾ 1 implies v(pc) ⩾ \frac{p}{p−1} − p^{(i−1)+1}α. It remains to notice that p^{2} ∈ ˜℘_{K} ⊂ Ξ(m + 1, α). □

Corollary 2.2.2. We have

Ξ(m, α) = { ∑_{j=m}^{∞} a_{j}^{p^{i(j)}}c_{j} | a_{j} ∈ O_{K} },

where i(j) = min\{ i ⩾ 0 | j/e_{k} ⩾ \frac{p}{p−1} − p^{i+1}α \}, and c_{j} are certain fixed elements of k with v_{k}(c_{j}) = j.

Proof. Induction on \frac{pe_{k}}{p − 1} − m. □

It is easy to see that Ξ(0, α) is a subring in O_{K}. Lemma 2.2.1 implies that any u ∈ Ξ(0, α) with v(u) = 0 is invertible in Ξ(0, α).

Definition. Let char K = 0. For any α ∈ ℚ, 0 ⩽ α ⩽ \frac{1}{p−1} we define E_{α} as the subgroup of K^{∗} which is equal to Ξ(0, α)^{∗} ·k^{∗}. We have obviously E_{0} = (1+ ˜℘_{K})·k^{∗} and E_{\frac{1}{p−1}} = K^{∗}. Next, E_{α} ⊂ E_{β} for α < β. For any u ∈ K^{∗}, denote w_{K}(u) = inf{ α | u ∈ E_{α} } = min{ α | u ∈ E_{α} }.

Remark. In the case ζ_{p} ∈ K the filtration (E_{α} · (K^{∗})^{p})_{α} coincides with the infernal part of the filtration on K^{∗}, dual (in the sense of Kummer duality) to the ramification filtration on the Galois group of the maximal abelian extension of K of exponent p introduced in [Z2]. The requirement of such compatibilty explains the “annoying” factor p^{i+1} in the definition instead of more natural p^{i}.

Lemma 2.2.3. Let a ∈ U_{K}, \overline{a} ∉ \overline{K}^{p}, c ∈ k, v_{k}(c) = j_{0} > 0, u ∈ K, v_{k}(u − 1 − a^{p^{i}}c) > j_{0}. Then w_{K}(u) ⩾ (\frac{p}{p−1} − \frac{j_{0}}{e_{k}})/p^{i+1}.

Proof. Let w_{K}(u) < (\frac{p}{p−1} − \frac{j_{0}}{e_{k}})/p^{i+1} which is equivalent to u ∈ E_{α} for some

(4)  α < \frac{\frac{p}{p−1} − \frac{j_{0}}{e_{k}}}{p^{i+1}} .

We have u − 1 ∈ Ξ(0, α), v_{k}(u − 1) = j_{0} whence u − 1 ∈ Ξ(j_{0}, α). Corollary 2.2.2 implies

u − 1 = ∑_{j=j_{0}}^{∞} a_{j}^{p^{i(j)}}c_{j},

where c_{j_{0}} = c, and c_{j} with j > j_{0} are arbitarily chosen. After division by c we obtain

a^{p^{i}} ≡ a_{j_{0}}^{p^{i(j_{0})}} mod 𝔐_{K},

and (ā)^{p^{i}} ∈ \overline{K}^{p^{i(j_{0})}}. In view of ā ∉ \overline{K}^{p}, we obtain i(j_{0}) ⩽ i, i. e. j_{0}/e_{k} ⩾ \frac{p}{p−1} − p^{i+1}α, contradicting (4). □

Lemma 2.2.4. Consider b_{0} = 1 + a^{p^{n}}c where 0 ⩽ n ∈ ℤ, a ∈ U_{K}, ā ∉ \overline{K}^{p}, c ∈ k and 0 < v(c) < \frac{p}{p−1}. Then

w_{K}(b_{0}) = \frac{\frac{p}{p−1} − v(c)}{p^{n+1}}.

Next, if b ∈ b_{0}(K^{∗})^{p}, then

w_{K}(b) ⩾ \frac{\frac{p}{p−1} − v(c)}{p^{n+1}}.

Proof. The first assertion follows immediately from the definition of w_{K} and Lemma 2.2.3. To prove the second assertion, we consider first the case when either n = 0 or p ∤ v_{k}(c). Suppose there exists b ∈ b_{0}(K^{∗})^{p} with w_{K}(b) < w_{K}(b_{0}). Among all such b we choose an element with v(b) = 0 and the maximal v(b − 1); we have b = b_{0}d^{p}, d ∈ U_{K}.

It can be seen from Lemma 2.2.3 that for v(d − 1) > v(b_{0} − 1)/p we have

w_{K}(b_{0}d^{p}) ⩾ \frac{\frac{p}{p−1} − v(c)}{p^{n+1}}.

The case v(d−1) = v(b_{0} −1)/p < \frac{1}{p−1} is possible only when p | v_{k}(c), and it follows n = 0. Since in this case \overline{(d^{p} − 1)/c} ∉ \overline{K}^{p}, Lemma 2.2.3 implies the same inequality for w_{K}(b_{0}d^{p}). Finally, consider the case v(d − 1) < v(b_{0} − 1)/p. If v(d − 1) > 0, we write d as (1+a_{0}^{p^{n_{0}}} c_{0})d′, a_{0} ∈ U_{K}, ā_{0} ∉ \overline{K}^{p}, c_{0} ∈ k and v(d′ −1) > v(d−1) = v(c_{0}); in the case v(d − 1) = 0 we write d = a_{0}^{p^{n_{0}}} d′ with the same conditions on a_{0} and d′. Then for v(d − 1) > 0 we have b = b_{0}d^{p} = 1 + a_{0}^{p^{n_{0}+1}}c_{0}^{p} + …, where dots denote terms of higher order. Lemma 2.2.3 implies

(5)  w_{K}(b) ⩾ \frac{\frac{p}{p−1} − pv(c)}{p^{n_{0}+2}}.

On the other hand, it can be immediately seen that

(6)  w_{K}((1 + a_{0}^{p^{n_{0}}} c_{0})^{p}) ⩽ \frac{\frac{p}{p−1} − pv(c)}{p^{n_{0}+2}}.

Denote b_{1} = b/(1 + a_{0}^{p^{n_{0}}} c_{0})^{p}. We observe from (5) and (6) that w_{K}(b_{1}) ⩽ w_{K}(b), whereas v(b_{1} − 1) > v(b − 1), a contradiction with the choice of b. In the case v(d − 1) = 0 a similar argument works for b_{1} = b/(a_{0}^{p^{n_{0}}} )^{p}.

Now consider the remaining case when n > 0 and p | v_{k}(c). Use induction on n. Consider any c_{1} ∈ k such that v(c − c_{1}^{p}) > v(c); we have, of course, v(c_{1}) = v(c)/p. Denote b′ = b_{0}/(1 + a^{p^{n}}c_{1}^{p}). An immediate calculation using Lemma 2.2.1 shows that

w_{K}(b′) > \frac{\frac{p}{p−1} − v(c)}{p^{n+1}}.

Similarly,

1 + a^{p^{n}}c_{1}^{p} = (1 − pa^{p^{n−1}}c_{1})(1 + a^{p^{n−1}}c_{1})^{p}b′′,

where

w_{K}(b′′) > \frac{\frac{p}{p−1} − v(c)}{p^{n+1}}

as well. We obtain b = b_{1}b′b′′, where b_{1} ∈ (1 − a^{p^{n−1}}pc_{1})(K^{∗})^{p}. By the assumption of induction,

w_{K}(b_{1}) ⩾ \frac{\frac{p}{p−1} − v(pc_{1})}{p^{n}} = \frac{\frac{p}{p−1} − 1 − \frac{v(c)}{p}}{p^{n}} = \frac{\frac{p}{p−1} − v(c)}{p^{n+1}},

which proves the second assertion of Lemma. □

Lemma 2.2.5. Let B be a class in K ∗ /(K^{∗})^{p}. Then there exists b ∈ B such that w_{K}(b) ⩽ w_{K}(b′), and v(b − 1) ⩾ v(b′ − 1) for any b′ ∈ B.

Proof. Similarly to Lemma 2.1.5. □

2.3. Infernal jump of a cyclic extension of degree p.

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

Definition. In the case char K = p by the infernal jump of L/K, where L = K(x), x^{p} − x = a ∈ K, we mean

w(L/K) = min{w_{K}(b) | b ∈ a + ℘(K)}.

In the case char K = 0 introduce K′ = K(ζ_{p}), L′ = L(ζ_{p}). Let L′ = K′(x), x^{p} = a ∈ K′. Then by the infernal jump of L/K we mean

w(L/K) = min{w_{K′}(b) | b ∈ a(K′^{∗})^{p}}.

If L/K is almost constant, we assume w(L/K) = 0.

Lemma 2.3.

1) We have w(L/K) ⩽ h(L/K);

2) If K′/K is any constant totally ramified extension of degree p, then w(L/K) = w(LK′/K′).

Proof. Let char K = p. It is clear from the definition of w_{K} that w_{K}(b) ⩽ −v(b)/p for any b ∈ K, and we deduce the first assertion by means of Proposition 1.4. In the case char K = 0 the argument is the same, if we start with w_{K}(u) ⩽ max(0, \frac{1}{p−1} −

\frac{v(u−1)}{p}) for u ∈ U_{K}.

To prove the second assertion, denote L′ = LK′. Let first char K = p. Assume that L/K is determined by the equation ℘(x) = a, where w(L/K) = w_{K}(a). The extension L′/K′ is determined by the same equation, whence w(L′/K′) ⩽ w_{K}(a) = w(L/K). It remains to prove that w(L′/K′) ⩾ w_{K}(a) = w(L/K).

Consider the expansion from Corollary 2.1.2:

a = ∑_{j=v(a)e_{k}}^{∞} a_{j}^{i(j)}c_{j}

with a_{j} ∈ U_{K} ∪ {0}. It is clear from Lemma 2.1.3 that w_{K}(a_{j}^{p^{i(j)}}c_{j}) ⩽ w_{K}(a) = w(L/K) for any j, and we have equality for some j. Take the minimal j_{0} such that the value of w_{K}(a_{j_{0}}^{p^{i(j_{0})}}c_{j_{0}}) is equal to w(L/K). Now we can assume without loss of generality that the value of j_{0} is maximal possible when a is running over elements of the same class modulo ℘(K) with the same value of w_{K}(a). This assumption implies that either i(j_{0}) = 0 or p ∤ v_{k}(c_{j_{0}}). It follows, in particular, that the equality w_{K}(a_{j}^{p^{i(j)}}c_{j}) = w_{K}(a_{j_{0}}^{p^{i(j_{0})}}c_{j_{0}}) is not possible for any j > j_{0}.

Thus, we have a = a_{j_{0}}^{p^{i(j_{0})}}c_{j_{0}} +b with w_{K}(b) < w_{K}(a). Next, w(L′/K′) = w_{K′}(a′) for some a′ ∈ a + ℘(d), d ∈ K′. By Lemma 2.1.4,

w_{K′}(a_{j_{0}}^{p^{i(j_{0})}}c_{j_{0}} + ℘(d)) ⩾ w_{K′}(a_{j_{0}}^{p^{i(j_{0})}}c_{j_{0}}) = w_{K}(a_{j_{0}}^{p^{i(j_{0})}}c_{j_{0}}) = w(L/K),

whereas w_{K′}(b) ⩽ w_{K}(b) < w(L/K). It follows w(L′/K′) ⩾ w(L/K).

In the case char K = 0 the argument is similar, if one uses the assertions 2.2.2– 2.2.4 instead of 2.1.2–2.1.4. □

2.4. On the composite of two cyclic p-extensions.

Lemma 2.4.1. Let L/K be a cyclic extension of degree p, h = h(L/K) > 0. Suppose that w(L/K) ⩽ h/p. Then for any c ∈ k there exists x ∈ L such that v(x^{p} − c) ⩾ v(c) + (p − 1)h.

Proof. First of all, notice that the assumptions h(L/K) > 0, w(L/K) < h(L/K) imply that L/K is totally ramified.

In the case char K = p we assume that the extension L/K is determined by the equation α^{p} −α = a where a has the maximal valuation in its class in K/℘(K), and w_{K}(a) = w(L/K). (Such a exists by Lemma 2.1.5.) In the case char K = 0 denote ρ = ζ_{p} − 1. We assume that the extension L/K is determined by the equation (1 + ρα)^{p} = 1 + ρ^{p}a where a is also chosen to have the maximal possible valuation and to satisfy w_{K}(1 + ρ^{p}a) = w(L/K). (Such a exists by Lemma 2.2.5.)

Let v_{K}(a) = −n, where n > 0, we have then n = pe_{k}h. We see immediately from the condition w(L/K) ⩽ h/p = n/(p^{2}e_{k}) that

(1)  a = ∑_{i=−n}^{−[n/p]−1} a_{i}^{p}π^{−i} + a_{0},

where a_{i} ∈ O_{K} for all −n ⩽ i ⩽ −[n/p] − 1, a_{0} ∈ K, v_{K}(a_{0}) ⩾ −[n/p].

Next, v_{K}(α) = −n/p. Let us except for a while the case when char K = 0 and h = \frac{1}{p−1}. We have then (n, p) = 1. There exist positive integers c and d such that cn − dp = 1. We see immediately that π′ = a^{−c}π^{−pd} is prime in K, and π_{1} = α^{−c}π^{−d} is prime in L. We have

π_{1}^{−p} − π′^{−1} = (α^{cp} − a^{c})π_{0}^{pd} = (α^{cp} − (α^{p} − α)^{c})π_{0}^{pd} = (cα^{p(c−1)+1} + …)π_{0}^{pd},

where dots denote terms of higher order. Therefore,

v_{K}(π_{1}^{−p} − π′^{−1}) = (p(c − 1) + 1) (−\frac{n}{p}) + pd = n (1 − \frac{1}{p}) − 1.

In the exceptional case it is well known that L = K(π_{1}), where π_{1}^{p} = π′ is a prime in K, and π_{1} is a prime in L.

For any m ⩽ n we can estimate the valuation of π_{1}^{−mp} − π′^{−m}:

v_{K}(π_{1}^{−mp} − π′^{−m}) = (−p(m − 1) + (n(1 − \frac{1}{p}) − 1)p)/p = −m + n\frac{p − 1}{p}.

In view of this observation, it remains to prove that there exists an expansion

c = ∑_{i=v_{k}(c)}^{v_{k}(c)+n−[n/p]−1} g_{i}^{p}π′^{−i} + g_{0},

where g_{0}, g_{i} ∈ K, v(g_{i}) ⩾ 0, v(g_{0}) ⩾ v(c) + (p − 1)h.

It follows from (1) that

(2)  π′ = ∑_{i=1}^{n−[n/p]} d_{i}^{p}π^{i} + d_{0} = f(π),

where all d_{i} ∈ K, v_{K}(d_{0}) > n − [n/p] and v_{K}(d_{i}) ⩾ 0. It follows that

π = f^{−1}(π′) = ∑_{i=1}^{n−[n/p]} c_{i}^{p}π′^{i} + c_{0},

where all c_{i} ∈ K, v_{K}(c_{0}) > n − [n/p] and v_{K}(c_{i}) ⩾ 0. Now it is sufficient to substitute this expression for π into any expansion of c into power series with coefficients from R_{k}. In the case char K = 0 one should also take into account that for any x, y ∈ O_{K} we have v(x^{p} + y^{p} − (x + y)^{p}) ⩾ 1 > (n − [n/p] − 1)/e_{k}. □

Proposition 2.4.2. Let K_{1}/K, K_{2}/K be cyclic extensions of degree p, h_{1} = h(K_{1}/K), h_{2} = h(K_{2}/K). Suppose that w(K_{1}/K) ⩽ h_{1}/p and w(K_{2}/K) ⩽ h_{2}/p.

1) If h_{1} < h_{2}, then

h(K_{1}K_{2}/K_{2}) = h_{1}/p, h(K_{1}K_{2}/K_{1}) = h_{1}/p + h_{2} − h_{1}.

2) If h_{1} = h_{2}, then

h(K_{1}K_{2}/K_{2}) = h(K_{1}K_{2}/K_{1}) ⩽ h_{1}/p.

Proof. We shall check both assertions in parallel assuming that h_{1} ⩽ h_{2}. Denote n = h_{1}e_{k}.

In the case char K = p assume that the extension K_{1}/K is determined by the equation β^{p} − β = b, where b has the maximal possible valuation in its class in K/℘(K), w(K_{1}/K) = w_{K}(b). (Such b exists by Lemma 2.1.5.) In the case char K = 0 denote ρ = ζ_{p} − 1. We assume that the extension K_{1}/K is determined by the equation (1 + ρβ)^{p} = 1 + ρ^{p}b where b is also chosen to have the maximal possible valuation and to satisfy w_{K}(1 + ρ^{p}b) = w(K_{1}/K). (Such b exists by Lemma 2.2.5.)

By the definition of w_{K}, the condition w(K_{1}/K) ⩽ h_{1}/p means that there exist b_{i} ∈ U_{K} ∪ {0} and c_{i} ∈ k with −n ⩽ i < −[n/p] and b′ ∈ K, v_{K}(b′) ⩾ −[n/p], such that

b = ∑_{−n⩽i<−[ \frac{n}{p}]} b_{i}^{p}c_{i} + b′.

Applying Lemma 2.4.1 to K_{2}/K, for any given i one can find c′_{i} ∈ K_{2} such that

v(c′_{i}^{p} − c_{i}) ⩾ v(c_{i}) + (p − 1)h_{2} ⩾ (p − 1)h_{1}.

In the case char K = p we conclude that

b ≡ b^{∗} = ∑_{−n⩽i<−[ \frac{n}{p}]} b_{i}c′_{i} + b′′ mod ℘(K_{2}),

where v(b′′) ⩾ (p − 1)h_{1}. Moreover, if h_{1} > h_{2}, then v(b′′) > (p − 1)h_{1}. (Take into account that v_{K}(b′) ⩾ −[n/p] > −n/p.) It follows

j(K_{1}K_{2}/K_{2}) ⩽ \frac{−v(b^{∗})}{p} ⩽ \frac{n}{p^{2}} = h_{1}/p.

If h_{1} < h_{2}, we obtain j(K_{1}K_{2}/K_{2}) = h_{1}/p. To calculate h(K_{1}K_{2}/K_{2}), apply Lemma 1.3.1.

The case char K = 0 can be done similarly. □

Proposition 2.4.3. Let K_{1}/K, K_{2}/K be cyclic extensions of degree p, h_{1} = h(K_{1}/K), h_{2} = h(K_{2}/K), w = w(K_{1}/K). Suppose that K_{2}/K is constant totally ramified, maybe inseparable, and w > h_{1}/p. If char K = 0, suppose that ζ_{p} ∈ k.

1. If w ⩾ h_{2}/p + h_{1} − h_{2}, then K_{1}K_{2}/K_{2} is ferociously ramified, and we have h(K_{1}K_{2}/K_{2}) = w.

2. If w < h_{2}/p + h_{1} − h_{2}, then h(K_{1}K_{2}/K_{2}) = h_{2}/p + h_{1} − h_{2}.

Proof. For simplicity, we treat only the case char K = p. Then K_{1}/K is determined by the equation ℘(x) = a, where w_{K}(a) = w, and −v(a)/p = h_{1}. (We can choose a suitable a by Lemma 2.1.5.) As in the proof of Lemma 2.3, we have

w_{K}(a) = w_{K}(a_{j_{0}}^{p^{i(j_{0})}}c_{j_{0}}) = \frac{−v(c_{j_{0}})}{p^{i(j_{0})+1}}

for some j_{0}, where all letters have the same meaning as in the proof of Lemma 2.3. In our case we have

\frac{−v(c_{j_{0}})}{p^{i(j_{0})+1}} > \frac{h_{1}}{p} = \frac{−v(a)}{p^{2}}.

Taking into account −v(c_{j_{0}}) ⩽ −v(a), we obtain i(j_{0}) = 0 and v(c_{j_{0}}) < v(a)/p. Therefore, w_{K}(a) = w_{K}(a_{j_{0}}c_{j_{0}}) = −v(c_{j_{0}})/p, and \overline{a_{j_{0}}} ∉ \overline{K}^{p}.

The case when w = h_{1} is obvious. We may therefore assume that w < h_{1}, i. e.

j_{0} < j_{1} = v_{K}(a). This means \overline{a_{j_{1}}^{p^{i(j_{1})}}} ∈ \overline{K}^{p}.

Suppose that w > h_{2}/p+h_{1}−h_{2}. Let k_{2} be the constant subfield in K_{2}. Applying Lemma 2.4.1 to k_{2}/k, for any j ⩾ j_{1} we can find an element c′_{j} ∈ k_{2} such that

v(c′_{j}^{p} − c_{j}) ⩾ je_{k} + (p − 1)h_{2} ⩾ −ph_{1} + (p − 1)h_{2} > −pw > −h_{1}.

It follows easily that

a = a_{j_{0}}c_{j_{0}} + b′ + ℘(b) = a′ + ℘(b)

for some b, b′ ∈ K_{2}, where v(b′) > −pw. Obviously, p|v_{K_{2}}(a′), and

v(a′) = −pw = max{v(a′′)|a′′ ∈ a + ℘(K_{2})}.

Proposition 2.1.1 implies that K_{1}K_{2}/K_{2} is ferociously ramified, and h(K_{1}K_{2}/K_{2}) = w.

In the case w = h_{2}/p + h_{1} − h_{2} we obtain

a = a_{j_{0}}c_{j_{0}} + b′ + a_{j_{1}}^{p^{i(j_{1})}}(c_{j_{1}} − c′_{j_{1}}^{p}) + ℘(b),

and v(c_{j_{1}} − c′_{j_{1}}^{p}) ⩾ −pw = v(a_{j_{0}}c_{j_{0}}). Here \overline{c_{j_{0}}^{−1}(a_{j_{0}}c_{j_{0}} + b′ + a_{j_{1}}^{p^{i(j_{1})}}(c_{j_{1}} − c′_{j_{1}}^{p}))} ∉ \overline{K}^{p}, and we continue as in the preceding case.

Finally, let w < h_{2}/p + h_{1} − h_{2}. (In particular, h_{2} < h_{1}.) Consider

ã = ∑_{j=j_{1}}^{j_{0}−1} a_{j}^{p^{i(j)}}c_{j}.

It is clear that w_{K}(ã) ⩽ \frac{−v(c_{j_{1}} )}{p^{2}} = h_{1}/p. Let K̃_{1}/K be the extension of K which corresponds to the Artin–Schreier polynomial X^{p} − X − ã. We have h(K̃_{1}/K) = h_{1} and w(K̃_{1}/K) ⩽ h_{1}/p. Applying Proposition 2.4.2 to K̃_{1}/K and K_{2}/K, we obtain h(K̃_{1}K_{2}/K_{2}) = h_{2}/p + h_{1} − h_{2}. This means that ã = a′ + ℘(b), b ∈ K_{2}, −v(a′)/p = h_{2}/p+h_{1} −h_{2}. It remains to notice that v(a−ã) = −pw > −p(h_{2}/p+h_{1} −h_{2}). □

§3. Elimination of ramification

Throughout this section K denotes a complete discretely valued field of any characteristic with the residue field of characteristic p, B a certain fixed base subfield in K, k the corresponding constant subfield in K, v = v_{B}.

3.1. On an infernal extension of degree p.

Proposition 3.1. Let K be standard, C/k a deeply ramified extension. If char K = 0, suppose that ζ_{p} ∈ k. Then for an infernal totally ramified Galois extension L/K of degree p the extension k′L/k′K is ferociously ramified for some finite subextension k′/k.

Proof. Consider a tower k ⊂ k_{0} ⊂ k_{1} ⊂ … ⊂ C, where each k_{i+1}/k_{i} is totally ramified of degree p, and k_{0}/k is finite tamely ramified. Denote K_{i} = k_{i}K, L_{i} = k_{i}L, h_{i} = h(K_{i+1}/K_{i}) = h(k_{i+1}/k_{i}), h′_{i} = h(L_{i}/K_{i}) for i ⩾ 0. The value of w(L_{i}/K_{i}) does not depend on i by Lemma 2.3. This common value will be denoted by w. Notice that w > 0 since L/K is infernal.

The definition of a deeply ramified extension implies that at least one of the following two conditions is satisfied.

(1) There exists infinitely many i such that k_{i+1}/k_{i} is inseparable. (In this case h_{i} = ∞.)

(2) We have ∑_{i} h_{i} = ∞, where the sum is taken over all i ⩾ 0 such that h_{i} ≠ ∞. Suppose that for any i the extension L_{i}/K_{i} is totally ramified. Then by Propo-

sitions 2.4.2 and 2.4.3 for any i we have either h′_{i+1} ⩽ h′_{i}/p, or h′_{i+1} = h′_{i} − \frac{p−1}{p}h_{i}. The first variant cannot occur infinitely many times since h′_{i} ⩾ w for all i. Therfore, there exists i_{0} such that h′_{i+1} = h′_{i} − \frac{p−1}{p}h_{i} for i ⩾ i_{0}. For any such i the extension k_{i+1}/k_{i} is separable and the series ∑_{i=i_{0}}^{∞} h_{i} does not converge. This also contradicts to the condition h′_{i} ⩾ w. □

3.2. Main theorem.

Theorem 3.2.1. Let K be a complete discretely valued field, k its constant subfield, C/k a certain deeply ramified extension. Then for any infernal extension L/K there exists a finite subextension k′/k in C/k such that k′L/k′K is ferociously ramified.

Lemma 3.2.2. Let L/K be a finite infernal extension, U/K a finite weakly ramified extension. If UL/U if ferociously ramified, then L/K if ferociously ramified as well.

Proof. Obvious. □

Proof of Theorem. 1. Reduction to the case of standard K.

Suppose that the assertion of Theorem is true in the case of standard K. Choose a standard K_{0} ⊂ K such that K/K_{0} is finite. (In the mixed characteristic case one can take any absolutely unramified subfield with the residue field \overline{K}. In the equal characterisic case we can take K_{0} to be the completed composite of B an K̃, where K̃ ≈ \overline{K} is any coefficient subfield in K.) Let K_{1}/K_{0} be the maximal almost constant subextension in K/K_{0}. Then K_{1} is almost standard, and K/K_{1} is infernal. Choose unramified U/K_{1} such that U is standard. Then both UK/U and UL/UK are infernal. Using our assumption, we obtain the existence of finite k′/k in C/k such that k′UK/k′U is ferociously ramified. Now it is clear that k′UK is standard. By Lemma 1.3.2, l/k′ is deeply ramified. Using once more our assumption, we obtain the existence of a finite subextension k′′/k′ in l/k′ such that k′′UL/k′′UK is ferociously ramified. By Lemma 3.2.2 k′′L/k′′K is ferociously ramified as well.

2. Reduction to the case [L : K] = p.

In L/K consider a chain of subextensions of degree p. Then one can apply consequently Lemma 1.3.2.

3. Reduction to the case [L : K] = p with normal L/K.

Let L/K be of degree p, not necessarily normal. Denote by M/K its normal closure, and by M_{0} the ramification subfield in M/K. Notice that [M : K] is a divisor of p!. Then it is clear that M/M_{0} is of degree p and normal, whereas M_{0}/K is weakly ramified. It follows that M_{0} is almost standard. Choose an unramified U/M_{0} and complete the argument similarly to part 1 of the proof.

4. Reduction to the case ζ_{p} ∈ K in the mixed characteristic case.

It is sufficient to apply Lemma 3.2.2.

5. The case [L : K] = p with L/K where K is standard, and ζ_{p} ∈ K in the mixed characteristic situation.

It is sufficient to apply Proposition 3.1. □

As a particular case, we obtain the following result which has in fact already appeared in the work of Hyodo.

Corollary 3.2.3. The assertion of Theorem 3.2.1 is true for any extension C/k with the Galois group ℤ_{p} assuming that C/k is not unramified.

Proof. Let C/k be a ramified ℤ_{p}-extension. Denote by k_{n} the intermediate field of degree p^{n} over k. Then k_{N+1}/k_{N} is ramified for some N, whence d_{k}(k_{N+1}/k_{N}) = ε > 0. Applying Hyodo inequality (see, e. g., [Z1, §1]), we can conclude that d_{k}(k_{n+1}/k_{n}) ⩾ ε for all n ⩾ N, whence C/k is deeply ramified. □

Remark. The requirement that C/k is deeply ramified is essential for Theorem 3.2.1. We give an example of an infinite extension C/k of a complete discretely valued field with perfect residue field, for which the conclusion of the theorem is not satisfied.

Let F be an arbitrary infinite perfect field of characteristic p, B = F ((π)) the base subfield. Let I be a basis of F as linear space over 𝔽_{p}. For any t ∈ I denote by x_{t} some root of the polynomial X^{p} − X − π^{−1}t, and put

C = B(x_{t} | t ∈ I).

Let K = F ((T))((π)), L = K(y), where y^{p} − y = π^{−2}T^{p}. Obviously, k = B is the constant subfield in K, and L/K is infernal. We shall show that for any finite subextension k′/k in C/k the extension k′L/k′K is not weakly unramified.

In fact, increasing k′/k, we may assume k′ = k(x_{t_{1}}, …, x_{t_{n}}), t_{1}, …, t_{n} ∈ I. Put k_{i} = k(x_{t_{1}}, …, x_{t_{i}}), i = 0, …, n. Then Proposition 2.1.3 implies j(k_{i}/k_{i−1}) ⩽ p^{−i}, i = 1, …, n. Hence also j(k_{i}K/k_{i−1}K) ⩽ p^{−i}, i = 1, …, n.

Next, we have w(k_{i}L/k_{i}K) = w(L/K) = 2p^{−2} for any i, whereas h(L/K) = 2p^{−1}. Applying Proposition 2.4.2, we obtain by induction on i that h(k_{i}L/k_{i}K) ⩾ p^{−1} + p^{−(i+1)}. It follows h(k_{n}L/k_{n}K) > w(k_{n}L/k_{n}K). This means that k_{n}L/k_{n}K is not weakly unramified.

3.3. On Epp–Ponomaryov theorem. We prove the following variant of the theorem of elimination of wild ramification (compare [E, Po2, Z1]). In certain sense this result permits one to strengthen the Epp–Ponomaryov theorem [Po2, Theorem 3]. However, we consider only finite extensions of complete fields here.

Theorem 3.3.1. Let L/K be any finite extension of complete discretely valued fields, k the constant subfield of the field K. Then there exists a finite solvable extension k′/k such that k′L/k′K is weakly unramified, i. e. e(k′L/k′K) = 1.

Denote by K_{0} any standard subfield in K such that K/K_{0} is a finite extension. Let e(L/K_{0}) = p^{h} · e_{t}, where (p, e_{t}) = 1. Then there exists a desired k′/k which is a composite of a cyclic p-extension and a certain extension of degree dividing d(p − 1)^{h}[L : K_{0}], where d = 1 in the case char K = p, and d = [K_{0}(ζ_{p}) : K_{0}] in the case char K = 0.

Remark. Take for K_{0} an absolutely unramified subfield in K such that K/K_{0} is totally ramified. Then we obtain Theorem 1 of Introduction.

Lemma 3.3.2. Let K be standard, L/K a weakly ramified extension, k the constant subfield in K. Then there exists a weakly ramified solvable extension k′/k such that [k′ : k] divides [L : K], k′ is the constant subfield in k′L, and k′L/k′K is unramified.

Proof. Without loss of generality we may assume that the constant subfield of L coincides with k. (If this is not so, we replace K with the maximal constant subextension in L/K.) Let L_{0} be the inertia subfield in L/K. Then L = L_{0}(\sqrt[e]{π_{0}}) where e = e(L/K), π_{0} is a prime in L_{0}. Denote k′ = k(\sqrt[e]{π}) where π is a prime in k. We have π_{0} = πu, v(u) = 0. Then k′L = k′L_{0}(\sqrt[e]{u}) is unramified over k′K. Since the normal closure of k′/k is k(ζ_{e}, \sqrt[e]{π})/k, k′/k is solvable. □

The following Lemma is due to F. Lorenz.

Lemma 3.3.3. Let L/K be a totally or ferociously ramified separable extension of degree p, L′/K the normal closure of L/K, E the ramification subfield in L′/K. Then L′/E is cyclic of degree p, and E/K is cyclic of degree dividing p − 1.

Proof. This is a general fact that [L′ : K]|p!. It follows immediately that L′/E is cyclic of degree p. Let G = Gal(L′/K), H = Gal(L′/L), G_{1} = Gal(L′/E).

Obviously, G_{1} is a normal subgroup in G. Since L′/K is the normal closure of L/K, the only subgroup of H which is normal in G is the trivial subgroup.

Now consider the action of H on G_{1} by conjugation. It yields a homomorphism

α: H → Aut(G_{1}) = ℤ/(p − 1)ℤ.

The elements of Ker α commute with the elements of G_{1}, and Ker α is a normal subgroup in H. Since H and G_{1} generate G, Ker α is normal in G as well, whence Ker α = id, and H is cyclic of degree dividng p − 1. It remains to notice that Gal(E/K) is isomorphic to H. □

Lemma 3.3.4. Let either char K = 0 and ζ_{p} ∈ K, or char K = p; L/K a totally ramified almost constant extension of degree p, k the constant subfield. Then there exists a solvable extension k′/k such that [k′ : k] | p(p − 1) and k′L/k′K is unramified.

Proof. Let L′/K be the normal closure of L/K, L_{1} the ramification subfield in L′/K. Then [L′ : L_{1}] = p, [L_{1} : K] | (p − 1) by the previous lemma. Let l_{1} be the constant subfield in L_{1}. It is clear from the properties 4 and 5 in 1.1 that for some cyclic k′/l_{1} of degree p the extension k′L′/k′L_{1} is unramified. Then k′L′/k′K is weakly ramified, and it follows that k′L/k′K is unramified since this is a pextension lying inside k′L′/k′K. It remains to note that [l_{1} : k] divides [L_{1} : K], whence [k′ : k] | p(p − 1). □

Proof of Theorem 3.3.1. Since K can be replaced by K_{0}, we may assume without loss of generality that K is standard.

Denote by L_{2}/K the maximal almost constant subextension in L/K. We prove first of all that there exists solvable k_{2}/k such that k_{2}L/k_{2}K is unramified, and [k_{2} : k] divides d(p − 1)^{h}[L : K], where d = 1 in the case char K = p and d = [K(ζ_{p}) : K] in the case char K = 0.

In this part of the proof we may increase K (if necessary) and assume that L/K has no non-trivial constant subextensions. In the case char K = 0 we may also assume that ζ_{p} ∈ K.

Let L_{1} be the ramification subfield in L_{2}/K. By Lemma 3.3.2 there exists a weakly ramified solvable extension k_{1}/k of degree dividing [L_{1} : K] such that k_{1}L_{1}/k_{1}K is unramified, and k_{1} is the constant subfield in k_{1}L_{1}.

The extension k_{1}L_{2}/k_{1}L_{1} is almost constant. It is totally ramified since L_{2}/L_{1} is a totally ramified p-extension, and k_{1}/k is weakly ramified. It is clear that its degree does not exceed p^{h}. Applying repeatedly Lemma 3.3.4, we get an extension k_{2}/k_{1} of degree dividing (p − 1)^{h}[L : L_{1}] such that k_{2}L_{2}/k_{2}L_{1} is unramified.

Now take any extension C/k with the Galois group ℤ_{p} which is not unramified. (It is easy to see that such an extension exists.) If k̃ is the constant subfield in k_{2}L_{2}, then k̃C/k̃ is also an extension with the Galois group ℤ_{p}, which is not unramified. At the same time k_{2}L/k_{2}L_{2} is infernal. By Corollary 3.2.3, in k̃C/k̃ there exists a finite subextension k̃_{3}/k̃ such that k̃_{3}L/k̃_{3}L_{2} is ferociously ramified. Next, it is obvious that k̃_{3} = k_{3}k̃ for some finite subextension k_{3}/k in C/k. The extension k_{3}/k is a cyclic p-extension, and this implies that k′ = k_{2}k_{3} is a desired one. □

3.4. Structure theorems. Here we prove slightly advanced versions of structure theorems for complete fields from [Z1].

Put e(K/k) = p^{h} · e_{t}, where (p, e_{t}) = 1.

Lemma 3.4.1. The field K is a totally ramified extension of a standard field K_{1} such that the constant subfield of K_{1} coincides with k, i. e. with the constant subfield of K.

Proof. If char K = 0, then by [FV, Prop. II.5.5 and II.5.6] there exists an absolutely unramified subfield K_{0} in K such that K/K_{0} is totally ramified. In the case char K = p, we assume K_{0} to be the completion of K̃B, where K̃ is any coefficient subfield in K, B the base subfield.

Thus, in both cases we observe a subfield K_{0} of K_{0} such that e_{K_{0}} = 1 and K/K_{0} is totally ramified. Then K_{1} = kK_{0} is the desired subfield. □

Theorem 3.4.2. There exists a constant solvable extension L/K such that L is a standard field. Moreover, L = L_{1}L_{2}, where L_{1}/K is a cyclic p-extension and [L_{2} : K] divides (p − 1)^{h+1}e(K/k). Furthermore, one may take L_{1} = lK, where l/k is a finite subextension of a given constant ℤ_{p}-extension C/k unless the latter is unramified.

Proof. Let K_{1} be as in Lemma 3.4.1. Then the application of Theorem 3.3.1 to K/K_{1} yields the required result. □

Corollary. Let K be a multidimensional local field of characteristic 0 with the first residue field of characteristic p > 0, the last residue field being arbitrary perfect. Then there exists a constant solvable extension L/K such that L is isomorphic to l{{t_{1}}} … {{t_{n}}}. Moreover, one may assume L = L_{1}L_{2}, where L_{1}/K is a cyclic p-extension, and [L_{2} : K] divides (p−1)^{h+1}e(K/k), where k is the constant subfield of K.

Theorem 3.4.3. Let k be the constant subfield in K, C/k a deeply ramified extension of k. Then there exists a finite subextension l/k in C/k such that lK is almost standard.

Proof. Choose K_{1} as in Lemma 3.4.1. Let K_{2}/K_{1} be the maximal almost constant subextension in K/K_{1}. By Theorem 3.2.1, in C/k there exists a finite subextension l/k such that lK/lK_{2} is ferociously ramified. Since K_{2} is almost standard, lK_{2} and lK are almost standard as well. □

Corollary. Let K be a multidimensional loacal field of characteristic 0 with the first residue field of characteristic p > 0, k its constant subfield. Suppose that C/k is a deeply ramified extension. Then there exists a finite subextension l/k in C/k such that lK is almost standard. In particular, there exists a semi-ramified^{2} extension L/K such that L(ζ_{p^{n}}) is a standard field for some positive integer n.

§4. On the choice of a constant subfield

It appears that when char K = p and the residue field is imperfect, one can force any totally ramified p-extension of K to be infernal by means of a suitable choice of a constant subfield. Namely, the following assertion takes place.

Proposition 4.1. Let K_{0} be a field of characteristic p with imperfect residue field, K/K_{0} a finite extension. Then for an arbitrary finite totally ramified Galois pextension L/K there exists a base subfield B ⊂ K_{0} such that extension L/K is infernal (with respect to B) and K_{0} is standard.

2i. e. unramified with respect to the rank 1 valuation

Proof. Let F be the maximal perfect subfield of \overline{K_{0}}. Choose a prime element π of K_{0} and a unit t ∈ K_{0} such that \overline{t} ∉ F. A base subfield B with the required properties can be found among the subfields of type B^{(N)} = 𝔽_{p}((πt^{N})), where N is an integer. Denote by k_{0}^{(N)} the completion of B^{(N)}(R_{K_{0}}), and by k^{(N)} the algebraic closure of k_{0}^{(N)} in K. Then k_{0}^{(N)} and k^{(N)} are the constant subfields with respect to B^{(N)} in the fields K_{0} and K respectively, and the field K_{0} is a fortiori standard.

Suppose that for any N the extension L/K is not infernal. Then for any N the extension L/K contains a non-trivial almost constant subextension L′/K. One can assume L′/K to be a Galois extension. Note that L′/K is p-extension and one can choose a normal subextension of degree p in L′/K. It appears to be an almost constant Artin-Schreier extension. We show the existence of N, for which any Artin-Schreier subextension in L/K is infernal. This yields a contradiction.

Consider the subextension K′/K in L/K, specified by the Artin-Schreier equation x^{p} − x = a. Since L/K is totally ramified, we have v_{K}(a) < 0 and we may assume p ∤ v_{K}(a). Suppose that the extension K′/K is an almost constant one (with respect to B^{(N)}). If necessary, the element a can be varied by an element of ℘(K), and one may assume a ∈ k^{(N)} +O_{K}. Thus a/c_{N} ∈ 1+𝔐_{K} for some c_{N} ∈ k_{N} and N_{K/K_{0}}a/N_{K/K_{0}}c_{N} ∈ 1 + 𝔐_{K_{0}}. The element c_{N} is algebraic over k_{0}^{(N)}, whence N_{K/K_{0}}c_{N} ∈ k_{0}^{(N)}.

Let n = v_{K_{0}}(N_{K/K_{0}}a). Then \overline{N_{K/K_{0}}a/(πt^{N})^{n}} ∈ F. Taking into account that, n ≠ 0 and t̄ ∉ F, we conclude that there exist at most one N, for which the extension K′/K is almost constant.

The extension L/K contains a finite number of subextensions of degree p, because there is a finite number of subgroups in the Galois group of L/K. Therefore one can choose some N = N_{0}, for which any such subextension is infernal. This completes the proof. □

Proposition 4.2. Let L/K be an arbitrary finite Galois extension and [L : K] = p^{r}s, where (s, p) = 1. Assume ζ_{s} ∈ K. There exists a base subfield B in K and a cyclic constant (with respect to B) extension K′/K such that e(K′L/K′) = 1.

Proof. Consider the tower of subextensions L/L_{1}/L_{0}/K corresponding to the filtration on the Galois group, where L_{0}/K is the maximal unramified subextension in L/K (inertia subfield), L_{1}/K the maximal tamely ramified subextension in L/K. Then L/L_{1} is a totally ramified Galois p-extension.

The previous proposition enables us to choose a base subfield B in K such that L/L_{1} becomes infernal and the field K standard. Let π ∈ B be a prime element of K.

The extension L_{1}/L_{0} is totally ramified Galois extension of degree that is prime to p. Let m = [L_{1} : L_{0}], (m, p) = 1. As m divides [L : K], then ζ_{m} belongs to L_{1}. There exists a prime element π_{0} ∈ K such that L_{1} = L_{0}( \sqrt[m]{π}_{0}) [FV, II.3.5]. Let K_{1} be the decomposition field for the equation x^{m} = π over K. The extension K_{1}/K is cyclic constant of degree m and L_{1}K_{1}/L_{0}K_{1} is unramified, because it is obtained by adjoining a root of degree m of π/π_{0} ∈ U_{L_{0}}.

Now we deal with the extension L/L_{1}. Consider an arbitrary extension C/k with the Galois group isomorphic to ℤ_{p}, that is not unramified. By Theorem 3.2.1 it contains a subextension k_{2}/k such that e(Lk_{2}/L_{1}k_{2}) = 1. Denote K_{2} = k_{2}K. Obviously, K_{2}/K is constant and cyclic, and [K_{2} : K] is a degree of p. Then K′ = K_{1}K_{2} is constant and cyclic, as a composite of two constant cyclic extensions with coprime degrees, and e(K′L/K′) = 1. □

Corollary 4.3. Let L/K be a separable subextension of a normal extension L′/K. Let [L′ : K] = p^{r}s, where (s, p) = 1. If ζ_{s} ∈ K, then there exist a base subfield B in K and a cyclic constant (with respect to B) extension K′/K such, that e(LK′/K′) = 1.

Corollary 4.4. If L/K is a Galois p-extension, there exist a base subfield B in K and a cyclic constant (with respect to B) extension K′/K such that e(K′L/K′) = 1.

Corollary 4.3 implies immediately

Theorem 4.5. Let ζ_{s} ∈ K for every s prime to p. For an arbitrary separable extension L/K there exist a base subfield B in K and a cyclic constant (with respect to B) extension K′/K such that e(K′L/K′) = 1.

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