On two approaches to classification of higher local fields1

This article links Kurihara’s classification of complete discrete valuation fields and Epp’s theory of elimination of wild ramification. For any complete discrete valuation fieldK with arbitrary residue field of prime characteristic one can define a certain numerical invariant Γ(K) which underlies Kurihara’s classification of such fields into 2 types: the field K is of Type I if and only if Γ(K) is positive. The value of this invariant indicates how distant is the given field from a standard one, i.e., from a field which is unramified over its constant subfield k which is the maximal subfield with perfect residue field. (Standard 2-dimensional local fields are exactly fields of the form k{{t}}.) We prove (under some mild restriction on K) that for a Type I mixed characteristic 2dimensional local field K there exists an estimate from below for [l : k] where l/k is an extension such that lK is a standard field (existing due to Epp’s theory); the logarithm of this degree can be estimated linearly in terms of Γ(K) with the coefficient depending only on eK/k.


Introduction
In the current paper we develop and compare two approaches to the classification of 2dimensional local fields in the mixed characteristic case. Here a 2-dimensional local field is a complete discrete valuation field such that its residue field has, in its turn, a structure of a complete discrete valuation field with perfect residue field of characteristic > 0.
If char = char , the field can be identified (non-canonically) with the field of formal Laurent series (( )). However, if char = 0 and char = , there is no explicit description and exhausting classification of such fields . Here are some known results in this direction.
First of all, there is an important subclass of such fields , so called standard fields. For any complete discrete valuation field with the residue field of characteristic > 0, one can introduce its constant subfield which is a maximal subfield of with perfect residue field. It can be proved that in the mixed characteristic case such is unique. The field is said to be standard if / = 1, where / is defined in 2.1.
This rather abstract definition working for any complete discrete valuation field with imperfect residue field, takes a very explicit form if is a 2-dimensional local field. Namely, if is standard and is its constant subfield, then conversely, if = {{ }} for a (one-dimensional) local field , then is standard, and is its constant subfield (see [8] or [14]). Note that in the very classical case, when the residue field of is finite, can be constructed as the maximal algebraic extension of Q inside .
Obviously, any 2-dimensional local field with local parameters ( , ) is a finite totally ramified extension of its standard subfield 0 = {{ }}, where is the constant subfield of , and , are as in 2.2 A non-trivial result following from Epp's theorem on elimination of wild ramification (see [1], [13]) is that for any such there exists a constant (i. e. defined over ) finite extension ′ / such that ′ is a standard field. In fact, there is a huge freedom in the choice of such ′ / , see [6]. However, the minimal degree ( ) of such ′ / can be arbitrarily large even in the simplest case [ : 0 ] = . Thus, ( ) seems to be an interesting invariant in the classification of 2-dimensional local fields.
Another approach to classification of mixed characteristic complete discrete valuation fields was initiated by Kurihara in [7] to study Milnor -groups (see [9] or [4]), These groups are applied in class field theory (see [10], [11], [4], [5]). Kurihara subdivides such fields into 2 types. For this, one considers any non-trivial relation · + · in the module of differentials of the given field over its constant subfield , where ( , ) are any local parameters of . The field belongs to Type I if ( ) < ( ) and to Type II otherwise (see [7], corollary 1.2 and definition 1.3). In particular, all standard fields belong to Type I since can be chosen from , and one can take = 1, = 0. Kurihara showed that the structure of extensions for the fields of Type I and Type II is very different. For example, has cyclic wild (resp. ferocious) -extensions of any degree if and only if is of Type I (resp. Type II). A refinement of this classification along with a number of new properties has been given in [2,3]. It was suggested to consider values like where is a second local parameter in a certain standard field containing , and the partial derivatives are used in the usual sense via identification = {{ }}. It is easy to see that so, the field is of Type I if and only if Δ( , ) > 0 for any choice of local parameters , . It can be shown that for the fields of Type I Δ( , ) does not depend on the choice of . For such fields, the value Γ( ) = sup is an invariant of measuring resemblance between and standard fields. In particular, Γ( ) = ∞ if and only if is "almost standard": a certain unramified extension of is a standard field. In this article we obtain a lower bound for ( ) for a mixed characteristic 2-dimensional local field of Type I, in terms of Γ( ) and ramification index of the field over its standard subfield. This is accomplished under a certain mild restriction on (Corollary 5.3.1). We are grateful to the referee of the first version of this article for valuable remarks.

Notation and basic definitions
The following notation is used throughout the paper: always denote a prime integer; ( ) is the -adic exponent of an integer number .

Discrete valuation fields
For a discrete valuation field , we denote its valuation by and its residue field by . For any such it will be always assumed that char = > 0. If char = > 0, we put = ( ). An element such that ( ) = 1 is said to be a uniformizer or .
Let / be an extension of valuation fields, be a valuation on , and induces the valuation on . We denote by / the index of ( * ) in ( * ). A finite extension / of discrete valuation fields is said to be • unramified, if / = 1, and / is separable; • tame, if / , and / is separable; • ferocious, if / = 1, and / is purely inseparable; • totally ramified, if / = | : |. By 0 we denote the valuation on any field normalized so that 0 ( ) = 1. For a Galois extension / of degree we denote by ( / ) the (Swan) ramification number of any generator of Gal( / ):

Two-dimensional local fields
Let be a two-dimensional local field; denote by (1) = its first residue field, and by (0) = (1) its last residue field. It is always assumed in this article that char = 0, char = > 0, and (0) is perfect.
Any two-dimensional mixed-characteristic local field K satisfies the conditions of 2.1. We will use the same notation, that is and 0 is such that For the valuation of rank 2 on we use notation = ( , ) : → Z 2 ; here Z 2 is linearly ordered as follows: ( , ) < ( , ), if < or = and < .
Given , we can define local parameters: a uniformizer with ( ) = (0, 1), and a "second local parameter" with ( ) = (1, 0). The constant subfield of is its maximal subfield such that its residue field (with respect to ) is perfect. In particular, if the last residue field of is finite, the constant subfield of is the algebraic closure of Q in .
In what follows denotes always a two-dimensional local field, and is its constant subfield. The field is said to be standard, if / = 1.
where is an algebraic extension of .

Kurihara's classification and related invariants
Let 0 = {{ }} be a standard 2-dimensional field. For ∈ 0 its formal derivative is defined as follows. If = ∑︀ with ∈ , then It is easy to see that is a well defined element of 0 .
Let 0 and 0 be standard fields with 0 ⊂ 0 , and let , ′ be second local parameters of these fields. Then ′ = ′ , where the first factor in the right hand side is the image in 0 of the respective element of 0 .
Let 0 be a standard field, a second local parameter of 0 , and , ∈ * 0 . Introduce Now we check that ( , ) is independent of the choice of 0 and the second local parameter 0 . Let 1 and 2 be standard fields with the second local parameters 1 and 2 , and let 1 ( , ) and 2 ( , ) be functions corresponding to these fields. There exists a standard field containing both 1 and 2 . Let be any second local parameter of . We have Note that for any , , we have In [2, 3] the notation Δ ( , ) was used for where , are local parameters of , and is a second local parameter of a standard field which is a finite extension of . In this article we redefine Δ ( , ) using 0 instead of , i. e., It is shown in [7,2] that if the condition Δ ( , ) > 0 is satisfied for some local parameters and of , then it is satisfied for any pair of local parameters. A field is of Type I if this condition is satisfied and is of Type II otherwise (see [2], proposition 4.3). For a field of Type I, Δ( , ) is independent of the choice of the second local parameter (see [2], Cor. 4.4); its value will be denoted by Δ ( ). Note that Then for any second local parameter of we have 3. Properties of ( ) 3.1 Proposition. Let , ∈ . Then: Proof. Direct calculation.
is finite, and denote an arbitrary number by otherwise. We claim that Γ ( ) . Let ′ be a common second local parameter of and ′ . Let ′ be a uniformizer of ′ such that ( ′ , ′ ) . Then = N ′ / ( ′ ) is a uniformizer of . Applying Corollary 4.1.1, we obtain Proof. The fields and ′ are of the same type by [7, Corollary 1.6].

Lemma. Let
Assume they are of Type I. Let / be the maximal unramified subextension in ′ / . Then / ′ is totally ramified. We will prove that Γ( ) = Γ( ), Γ( ′ ) = Γ( ). It is sufficient to check the inequalities: Denote by and arbitrary second local parameters of and . Then is also a second local parameter of ′ . We will prove that ( , ) = 0.
2) Now we prove Γ ( ) Γ ( ′ ), Γ ( ) Γ ( ). In view of 5, it is sufficient to prove that for any uniformizer of there exist uniformizers and ′ of and ′ such that ( , ) 0 and ( ′ , ) 0. Let be either In both cases we have ∈ and 0 ( ) = . Let ,1 , be arbitrary local parameters of , and let ∈ (0) , 1 ∈ Z, ∈ (1) be such that ,1 = [ ] 1 . Denote by a second local parameter of any standard field which is a finite extension of ′ , and denote by any integer number with 0 ( 1 − ) 0 (︀ )︀ ; put = 1 − . We will prove that the uniformizer Taking into account In the case = ′ we obtained the desired inequality, whereas in the case = it is a consequence of the above formula and Corollary 4.1.1.
Next, finite totally ramified Galois -extensions 1 / and 2 / are not in touch, if for any intermediate fields where / is normal and / is a Galois extension of degree ( = 1, 2), the extensions 1 2 / 1 2 and 1 2 / 1 2 are not in touch.
The idea behind this notion is that we consider extensions "in general position"such that the ramification of their compositum can be computed in terms of ramification of the original extensions, compare [12, 4.3].
We say that an extension ′ / is constant free, if ′ / is not in touch with any constant extension of . For example, for = {{ }}, where contains a primitive th root of unity, a Kummer extension (︀√︀ 1 + )︀ / with ∈ is constant free iff / ∈ .
5.1 Lemma. Let 1 / and 2 / be Galois extensions of degree that are not in touch. Assume that 1 2 / is totally ramified. Then: Proof. The first part follows immediately from Lemma 3.3.1 in [12]. (It is assumed there that the residue field is perfect but the proof goes through assuming only that 1 2 / is totally ramified.) For the second part it is sufficient to notice that ( 2 / ) < −1 . Indeed, if and 1 2 / is totally ramified, then 1 / and 2 / are always in touch; this can be seen from the explicit form of Kummer equations (after adjoining a primitive th root of unity).
Afterword Thus, we have established, under some restrictions, a relation between two invariants measuring how far is a given 2-dimensional local field from being standard. We expect that this relation, in some refined form, can be extended to all higher local fields.