{'fraction_non_alphanumeric': 0.07167032394837654, 'fraction_numerical': 0.009856064268977572, 'mean_word_length': 3.598597571404139, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Counting the number of distinct distances of elements in valued field extensions

The defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields, whenever positive characteristic is involved. We continue the detailed study of defect extensions through the tool of distances, which measure how well an element in an immediate extension can be approximated by elements from the base field. We show that in several situations the number of essentially distinct distances in fixed extensions, or even just over a fixed base field, is finite, and we compute upper bounds. We apply this to the special case of valued functions fields over perfect base fields. This provides important information used in forthcoming research on relative resolution problems.


Introduction
By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. The valuation ring of v on L will be denoted by O L , and that on K by O K . The value group of (L, v) will be denoted by vL, and its residue field by Lv. The value of an element a will be denoted by va, and its residue by av.
The defect, also known as ramification deficiency, of finite extensions (L|K, v) of valued fields is a phenomenon that only appears when the residue field Kv has positive characteristic. It is a main obstacle to the solution of deep open problems in positive characteristic, such as: Both problems are linked through the structure theory of valued function fields, in which it is essential to tame the defect, as well as wild ramification, cf. [7,10,12,13]. While implicitly known through the work of algebraic geometers and model theorists since the 1950s, the connection of the defect with the problem of local uniformization and the model theory of valued fields with positive residue characteristic has been pointed out in detail in the cited works of the second author. Defects also appear in crucial examples, as in the paper [3].
Using tools of ramification theory, the study of extensions of valued fields of residue characteristic p > 0 with nontrivial defect can be reduced to the study of normal extensions of degree p with nontrivial defect. Such extensions are immediate. An arbitrary extension (L|K, v) of valued fields is immediate if the canonical embeddings of vK in vL and of Kv in Lv are onto. As a consequence, for every a ∈ L \ K the set v(a − K) := {v(a − c) | c ∈ K} does not have a maximal element; this follows from [6,Theorem 1]. If a is an element of any valued field extension of (K, v) such that v(a − K) has no maximal element, then this set is an initial segment of vK. We associate with it a cut in the divisible hull vK of vK by taking as the lower cut set the smallest initial segment in vK which contains v(a − K). This cut is called the distance of a over K and denoted by dist (a, K). For more details, see Section 2.2.
Distances can be used to classify defect extensions. If an extension L|K of degree p is Galois and the field K is itself of characteristic p, then L|K is an Artin-Schreier extension, that is, L is generated over K by an element ϑ such that (1) ϑ p − ϑ ∈ K ; we call ϑ an Artin-Schreier generator of the extension. If such an extension of a valued field (K, v) has nontrivial defect, then the extension of the valuation v from K to L is unique and (L|K, v) is immediate (see Lemma 2 below); we call it an Artin-Schreier defect extension. A classification of Artin-Schreier defect extensions is introduced in [9], and it is shown that the classification can be read off from the distance dist (ϑ, K) of the Artin-Schreier generator. In a collaboration of the second author with O. Piltant ( [16]) the question arose how many distinct distances of generators of Artin-Schreier defect extensions exist over a fixed (K, v) (in particular, whether this number is finite at all).
, which means that the cut dist (a, K) is just shifted by adding vc to all elements of the lower cut set; we then write (2) dist (ca, K) = vc + dist (a, K) .
We do not regard dist (a, K) and dist (ca, K) as essentially distinct, so we will actually ask for the number of distances that are distinct modulo vK. In Section 4 we give an answer under certain finiteness assumptions, see Theorem 23. These conditions hold for instance in the following situation: Theorem 1. Take a valued function field (K|K 0 , v) over a perfect trivially valued base field (K 0 , v). Then the number of distinct distances of elements in Artin-Schreier defect extensions modulo vK is bounded by 2 · trdeg K|K 0 .
This answers a question from Olivier Piltant; results of this type are a crucial tool in [16].
More generally, we would like to count all the essentially distinct distances over K of all elements a ∈K for which v(a − K) has no maximal element. But it seems unlikely that we will get a finite number if we allow the elements a to attain arbitrarily large degree over K, so we need again some conditions. The first way to impose suitable conditions is to restrict the scope to all elements a ∈ L where L|K is a finite extension such that the extension of v from K to L is unique. For this case, we obtain in Section 3 an upper bound in terms of the defect of the extension (L|K, v) and its ramification index (vL : vK), see Theorem 19.
Another approach is to limit the scope to all a ∈K of bounded degree over K.
It is an open problem whether the number of essentially distinct distances in this case is always finite and to compute an upper bound for it, even under the finiteness conditions of Theorem 23. However, we are able to show that under these finiteness conditions, the number of distances that are distinct modulo vK is always finite; we give an upper bound in Theorem 24.
Note that there are examples of valued fields of rank 1, but infinite p-degree, where even the number of distances of elements in immediate purely inseparable extensions of degree p (and of elements in Artin-Schreier defect extensions) that are distinct modulo vK is infinite.
2.1. Defect. Take a finite normal extension L|K and a valuation v on K. Then v has finitely many distinct extensions v 1 , . . . , v g to L. All of them have the same ramification index (v i L : vK), which we will denote by e, and all of them have the same inertia degree [Lv i : Kv], which we will denote by f . Then we have the fundamental equality where by the Lemma of Ostrowski (cf. [ If L is a normal extension of prime degree p of (K, v) with nontrivial defect, then the extension of v from K to L is unique and (L|K, v) is immediate.
We will almost always consider extensions (L|K, v) for which the extension of v from K to L is unique. We will call such extensions uv-extensions in short; they are necessarily algebraic extensions. Note that every purely inseparable algebraic extension is a uv-extension.
For a finite uv-extension (L|K, v), we can define its defect even if the extension is not normal: A valued field (K, v) is henselian if it satisfies Hensel's Lemma, or equivalently, if the extension of v to the algebraic closureK of K is unique (i.e.,K is a uvextension of (K, v)). In this case, v extends uniquely to each algebraic extension of K. Every algebraically closed valued field is trivially henselian. Every valued field (K, v) admits a henselization, that is, a minimal henselian extension of (K, v), in the sense that it admits a unique valuation preserving embedding over K in every other henselian extension of (K, v). In particular, if w is any extension of v toK, then (K, v) has a unique henselization in (K, w), as it is the decomposition field of the normal extension (K sep |K, v), where K sep ⊆K is the separable-algebraic closure of K.
Henselizations of (K, v) are unique up to valuation preserving isomorphism over K. Moreover, they are always immediate separable-algebraic extensions of (K, v) (cf. [4,Theorem 17.19] For the remainder of this paper, we fix an extension of v from K tõ K. This will also fix the henselization of (K, v). Therefore, we will speak of the henselization of (K, v), and denote it by (K h , v).
Since the henselization is an immediate extension and the compositum L.K h of L and K h lies in L h (in fact, it is equal to L h ), this lemma yields:

2.2.
Distances. Take an arbitrary extension (L|K, v) of valued fields and a ∈ L\K. There are several possible definitions for the distance of a from K that have been used in papers by the first author. We choose the definition that is most suitable for our purposes in this paper.
By dist (a, K) we denote the cut induced by the set v(a−K)∩ vK in the divisible hull vK of vK. Namely, the lower cut set of dist (a, K) is the smallest initial segment that contains v(a − K) ∩ vK. This definition is slightly different from the one introduced in [9] and [17]. There, we have used the cut in vK induced by the subset v(a − K) ∩ vK to define dist (a, K). A detailed study of the new notion of distance and a comparison with the former notion can be found in [1]. Note that when v(a − K) ⊆ vK, the two notions coincide.
Our definition enables us to compare dist (a, K) with dist (a, L) when (L|K, v) is an algebraic extension since then, both dist (a, K) and dist (a, L) are cuts in the same ordered abelian group vK = vL. Then dist (a, K) < dist (a, L) will mean that the left cut set of dist (a, K) is a proper subset of that of dist (a, L).
The following is Lemma 3.9 of [1].
If (L|K, v) is an arbitrary valued field extension and a ∈ L, then we will say that a is weakly immediate over K if v(a − K) has no maximal element. In the language of pseudo Cauchy sequences, this means that a is a pseudo limit of a pseudo Cauchy sequence (also called "pseudo convergent sequence" in [6]) in (K, v) that has no pseudo limit in K. In the language used in [17] it means that the approximation type of a over K is immediate. Note that this does not imply that the extension (K(a)|K, v) is immediate (cf. [1,Example 3.17]). But conversely, by what we have already said in the introduction, every element in an immediate extension of (K, v) is weakly immediate over K. Observe that if a is weakly Lemma 7. Take a finite defectless uv-extension (L|K, v). Then the following as- Proof. a): This follows from Proposition 3.12 and Lemma 3.10 of [1]. b): This is Corollary 3.11 of [1].
To obtain another important distance equality, we need the following theorem from [11]: Then K h and K(a) are not linearly disjoint over K.
Lemma 9. Take an algebraic uv-extension (L|K, v). Then for all a ∈ L \ K which are weakly immediate over K, . But then by Theorem 8, K(a)|K and hence also L|K is not linearly disjoint from K h |K, a contradiction to Lemma 4. So we have that v(a − K) = v(a − K h ), which implies the equality of the distances.

2.3.
Weakly and strongly immediate elements. We have already defined what it means for an element in an extension of (K, v) to be weakly immediate over (K, v). A useful stronger property is the following. Take any extension (L|K, v) of valued fields and an element a ∈ L \ K. Then we will say that a is strongly immediate over K if v(a − K) has no maximal element and in addition, for every polynomial Proof. For the first assertion, see [17,Lemma 5.3]. The second assertion follows from the first together with Lemma 3.
In general, even if (K(a)|K, v) is a uv-extension and a is weakly immediate over K, the extension may not be immediate and a may not be strongly immediate over K. But this holds if the degree [K(a) : K] is a prime: Lemma 11. Take a uv-extension (K(a)|K, v) of prime degree p with its generator a weakly immediate over K. Then (K(a)|K, v) is immediate and a is strongly immediate over K.
is fixed for all such c. This contradicts our choice of f and shows that a polynomial g as chosen in the beginning cannot exist.
Lemma 12. Take a henselian field (K, v) and an element a ∈K which is weakly immediate over K. If a is not strongly immediate over K, then there is an imme- Proof. Using the notions of [17], we argue as follows. Since v(a−K) has no maximal element, the approximation type appr (a, K) is immediate by [17, Lemma 4.1 a)]. Take g to be an associated minimal polynomial for appr (a, K). Since the extension (K(a)|K, v) is not strongly immediate, we have that deg g < [K(a) : K]. Take b ∈K to be a root of g. Then [17,Theorem 6.4] shows that there is an extension w of v from K to K(b) such that (K(b)|K, w) is immediate and appr (b, K) = appr (a, K). Since (K, v) is henselian, w and v must agree on K(b), showing that (K(b)|K, v) is immediate. The equality of the approximation types implies that v(b − K) = v(a − K), which in turn implies that dist (b, K) = dist (a, K).

2.4.
The ramification field. For general ramification theory, see [5] or [15]. For information on tame valued fields, see [14]. We will summarise here the main properties of the ramification field that we will use.
Let (N |K, v) be a normal algebraic extension of henselian fields. We take the ramification field V of this extension to be the fixed field of the ramification group {σ ∈ Aut(N |K) | 0 = x ∈ O L ⇒ v(σx − x) > vx} of the automorphism group of N |K in the maximal separable subextension of N |K.
The absolute ramification field of a henselian field (K, v) is the ramification field of the normal algebraic extension (K sep |K, v), where K sep denotes the separable-algebraic closure of K.
Lemma 13. Take a normal extension (N |K, v) of henselian fields with residue characteristic p > 0. Then its ramification field V has the following properties: a) The extension V |K is separable.

b) Every subextension of N |V is a tower of normal extensions of degree p. c) The valued field extension (V |K, v) is tame and hence every finite subextension
e) For all a ∈ N \ K weakly immediate over K, dist (a, V ) = dist (a, K) .
Proof. Assertion a) follows from our definition.
Assertion b) follows from the fact that the ramification group is a p-group (cf. For assertion c), note that V is a subfield of the absolute ramification field K r of (K, v), which by part b) of [14, Lemma 2.13] is a tame extension of (K, v). Hence by part a) of the same lemma, also V is a tame extension of (K, v). Thus every finite subextension (E|K, v) of the tame extension (V |K, v) is defectless. In view of this, the equality of the defects follows from [9, Proposition 2.8].
For the proof of d) suppose that dist (a, V ) > dist (a, K). Then by Lemma 6 there is an element b ∈ V such that dist (a, K) = dist (b, K). On the other hand, (K(b)|K, v) is a defectless uv-extension, by part c). Together with part a) of Lemma 7 this contradicts the fact that a is weakly immediate over K.

The number of distinct distances in a given valued field extension
Take a finite (not necessarily immediate) uv-extension (L|K, v). We wish to count the number of distances appearing in this extension that are distinct modulo vK. We define ndd (L|K, v) to be the minimal m ≥ 0 such that there are elements a 1 , . . . , a m ∈ L \ K so that each a i is weakly immediate over K and for every b ∈ L \ K for which v(b − K) has no maximal element, there is i ∈ {1, . . . , m} and α ∈ vK with Proof. Assume first that ndd (L 0 |K, v) = 0 and take a ∈ L \ K weakly immediate over K. If dist (a, K) = dist (a, L 0 ), then dist (a, L 0 ) > dist (a, K) and by Lemma 6 there is b ∈ L 0 such that dist (a, K) = dist (b, K). But then v(a − K) has no maximal element, contradicting our assumption that ndd (L 0 |K, v) = 0. Now assume that ndd (L 0 |K, v) > 0 and take a ∈ L 0 \ K weakly immediate over K. Since a ∈ L 0 , it follows that dist (a, L 0 ) > dist (a, K).
Lemma 15. Take a finite uv-extension (L|K, v) and an algebraic extension (K ′ |K, v) such that (vK ′ : vK) < ∞ and dist (a, K) = dist (a, K ′ ) for all a ∈ L. Then Proof. Set n = (vK ′ : vK) and choose representatives β 1 , . . . , β n ∈ vK ′ of the distinct cosets in vK ′ /vK. If two distances dist (a 1 , K) and dist (a 2 , K) are equal modulo vK ′ then there is i ∈ {1, . . . , n} and α ∈ vK such that dist (a 1 , K) = α + β i + dist (a 2 , K), where the latter is equal to β i + dist (a 2 , K) modulo vK. This shows that the maximum number of distances that are distinct modulo vK but equal modulo vK ′ is n, which proves the first inequality.
The second inequality follows from the fact that all β i lie in vK.
The next lemma computes ndd (K(a)|K, v) for uv-extensions (K(a)|K, v) with a strongly immediate over K. We derive it from [  and h = p ℓ with 0 ≤ ℓ < k such that for all c ∈ K with v(a − c) ≥ γ, the value vf i (c) is fixed for each i ≥ 0, Therefore, ndd (K(a)|K, v) ≤ k and, modulo vK, all distances are multiples of dist (a, K) by powers of p.
Proof. Using the notions of [17], the assumption that a is strongly immediate over K is equivalent to the approximation type of a over The following corollary shows that a uv-extension of prime degree generated by a weakly immediate element admits exactly one distance modulo vK. It follows from the previous lemma together with Lemma 11. Corollary 17. Take a uv-extension (K(a)|K, v) of prime degree p such that a is weakly immediate over K. Then for every nonconstant polynomial f ∈ K[X] of degree smaller than p there is γ ∈ v(a−K) such that for all c ∈ K with v(a−c) ≥ γ, the value vf i (c) is fixed for each i ≥ 0, and Hence for any b ∈ K(a) \ K, Now we assume that for some i < n we have already shown that ndd (L i |K, v) ≤ m i · (vL i : vK) and ndd (L i |K, v) ≤ m i . Take any a ∈ L i+1 \ L i which is weakly immediate over K. Since [L i+1 : L i ] is prime, we have that L i+1 = L i (a). By Lemma 6, either dist (a, K) = dist (b, K) holds for some b ∈ L i , or dist (a, K) = dist (a, L i ).
Suppose that there is such an element a for which the latter holds. Then a is weakly immediate over L i and by Lemma 11, This says that modulo vL i , all distances dist (a, K) arising in this way must be equal. Consequently, there can be at most (vL i : vK) many that are distinct modulo vK, and only one modulo vK. This is in addition to the number of distinct distances arising from elements in L i . So we obtain that Suppose now that there is no such element a. Then This completes our induction.
We will now generalize this result to arbitrary finite, not necessarily immediate, uv-extensions. Proof. First, we show that we may assume (K, v) to be henselian. For every a ∈ L \ K, Lemma 9 shows that dist (a, K h ) = dist (a, K). By Lemma 15 we obtain that ndd ( Since the henselization of a valued field is an immediate extension of the field, (vL.K h : vK h ) = (vL h : vK h ) = (vL : vK). Thus, we may replace K by its henselization.
We denote the normal hull of L over K by N . Since (K, v) is henselian, there is a unique extension of v from L to N and (N |K, v) is again a uv-extension. Now we take V to be the ramification field of (N |K, v). From Lemma 13 we obtain that (V |K, v) is a defectless uv-extension such that d(L.V |V, v) = d(L|K, v) = p m and that dist (a, V ) = dist (a, K) for every a ∈ N \ K which is weakly immediate over K.

The number of distinct distances in all Artin-Schreier defect extensions
Throughout this section, let (K, v) be a field of positive characteristic p. As before, we assume that v is extended to the algebraic closureK of K. By Zorn's Lemma, there always exists a maximal immediate subextension (K ′ |K, v) of the purely inseparable (K 1/p |K, v), where K 1/p = {c 1/p | c ∈ K}. Throughout the present and the final section of this paper, we will assume that K ′ |K is finite, so that its degree is p m for some m ≥ 0. If K has finite p-degree k, that is, [K 1/p : K] = [K : K p ] = p k with k ≥ 0, then m ≤ k.
We will now apply our previous results to consider the possible distances (modulo vK) of all elements that are contained in any Artin-Schreier defect extension of (K, v). In view of Corollary 17, we only have to determine the distance of one generator of such an extension. The Artin-Schreier defect extension (K(ϑ)|K, v) with Artin-Schreier generator ϑ is called dependent if there is a purely inseparable immediate extension (K(η)|K, v) of degree p such that This implies that v(ϑ − c) = v(η − c) for all c ∈ K and that dist (ϑ, K) = dist (η, K) .
We note that by assumption, η ∈ K 1/p . Proposition 20. Under the assumptions on (K, v) outlined above, ndd ( Proof. For every a ∈ K 1/p which is weakly immediate over K, there must be some b ∈ K ′ with dist (a, K) = dist (b, K). Otherwise, we would obtain that dist (a, K ′ ) = dist (a, K) which yields that a is weakly immediate over K ′ ; since [K ′ (a) : K ′ ] = p, this would show by Lemma 11 that (K ′ (a)|K ′ , v) and hence also (K ′ (a)|K, v) are immediate extensions, contradicting the maximality of K ′ . So we have that ndd (K 1/p |K, v) = ndd (K ′ |K, v). Since For the proof of the last assertion, note that if (K, v) is of finite p-degree, then Thus m ≤ s. In order to make a statement about all possible Artin-Schreier defect extensions (K(a)|K, v), we also have to consider the independent ones, that is, the ones that are not dependent. It is shown in [9] that if a is an Artin-Schreier generator of the extension, then dist (a, K) is the lower edge of some proper convex subgroup H of vK, that is, the lower cut set of dist (a, K) is the largest initial segment of vK that does not meet H. We summarize: The rank of (K, v), if finite, is the number of proper convex subgroups of the value group vK. Putting the previous results together, we obtain: Theorem 23. Take a valued field (K, v) of finite rank r, satisfying the assumptions outlined in the beginning of this section. Then the number of distinct distances modulo vK of elements in all normal defect extensions of prime degree of (K, v) as well as the number of distinct distances modulo vK of elements in Artin-Schreier defect extensions of (K, v) are bounded by r + m. In particular, if K has finite p-degree k, then this number is bounded by r + k.
For a function field K over a perfect base field K 0 , the p-degree k is equal to the transcendence degree trdeg K|K 0 . For a valued function field (K|K 0 , v) over a trivially valued base field (K 0 , v), the rank is bounded by trdeg K|K 0 . This proves Theorem 1.

The number of distinct distances of all elements of bounded degree
Throughout this section we shall work under the following assumptions, unless indicated otherwise. We take (K, v) to be a valued field of positive characteristic p and finite rank r.
For every natural number i we denote by ndd * i (K, v) the number of distinct distances modulo vK of elements a ∈ K \ K satisfying the following conditions: We will show now that for every i ∈ N the number ndd * i (K, v) is finite. Theorem 24. Assume additionally that (K, v) has finite p-degree and d(K 1/p |K, v) = p m . Then ndd * i (K, v) is finite for every natural number i. More precisely, Proof. In what follows, let a ∈ K satisfy the assumptions (10). Lemma 9 shows that dist (a, K) = dist (a, K h ). This implies in particular that a is weakly immediate over K h . Furthermore, the assumptions (10) . We wish to show that also (K h , v) satisfies the assumptions stated at the beginning of this section. Since K h |K is a separable algebraic extension, K h has the same p-degree as K, so [(K h ) 1/p : K h ] = p k . It follows that (K h ) 1/p = K 1/p .K h . Since K 1/p |K is finite and v extends uniquely from K to K 1/p , Lemma 5 yields that . Furthermore, vK h = vK is again of rank r. Hence we can assume that (K, v) is henselian.
Take K r to be the absolute ramification field of K with respect to the fixed extension of v toK. Lemma 13 shows that dist (a, K) = dist (a, K r ). This implies in particular that a is weakly immediate over K r . Moreover, [K r (a) : K r ] ≤ [K(a) : K] and vK = vK r . Therefore, ndd * i (K, v) ≤ ndd * i (K r , v) for every i ∈ N. We wish to show that also (K r , v) satisfies the assumptions stated at the beginning of this section. Since K r |K is a separable algebraic extension, K r has the same p-degree as K, so [(K r ) 1/p : K r ] = p k . It follows that (K r ) 1/p = K 1/p .K r . Lemma 13 yields that p m = d(K 1/p |K, v) = d(K 1/p .K r |K r , v) = d((K r ) 1/p |K r , v).
Furthermore, vK r /vK is a torsion group, hence vK r is again of rank r. Hence we can assume that K r = K. Note that by Lemma 13 this means that the extension K(a)|K is a tower of normal extensions of degree p. In particular, it is of degree p t for some t ∈ {0, . . . , i}.
We proceed by induction on i. The case of i = 1 is covered by Theorem 23. Now assume that i ≥ 2 and ndd * i−1 (K, v) ≤ r + (i − 1)m .
To give an upper bound for ndd * i (K, v), it is enough to consider elements of degree p i over K which are weakly immediate over K. Indeed, the distances of elements a of degree at most p i−1 are already counted in ndd * i−1 (K, v). Hence we assume that [K(a) : K] = p i .
If a is not strongly immediate over K, then by Lemma 12 there is an immediate extension (K(b)|K, v) with dist (b, K) = dist (a, K) and [K(b) : K] < [K(a) : K]. By Lemma 13 the degree [K(b) : K] must be a power of p. We conclude that [K(b) : K] ≤ p i−1 , showing that dist (a, K) is already counted in ndd * i−1 (K, v). Hence we assume that a is strongly immediate over K. By Lemma 10 this implies that the extension (K(a)|K, v) is immediate.
Assume first that K(a)|K is purely inseparable. Then from Lemma 6 we deduce that dist (a, K) = dist (a, K 1/p i−1 ) or dist (a, K) = dist (d, K) for some d ∈ K 1/p i−1 . If the latter holds, then d is weakly immediate over K and therefore, dist (b, K) appears already as a distance of some weakly immediate element of degree ≤ p i−1 . So we may assume that the former holds. Then K 1/p i−1 (a)|K 1/p i−1 is a purely inseparable extension of degree p and the element a is weakly immediate over K 1/p i−1 . Since d(K 1/p i |K 1/p i−1 , v) = d(K 1/p |K, v) = p m , Proposition 20 shows that there are at most m distinct distances of elements of K 1/p i weakly immediate over K 1/p i−1 , modulo vK 1/p i−1 = 1 p i−1 vK, hence also modulo vK. This renders at most m additional distinct distances dist (a, K) modulo vK.
Assume now that K(a)|K is not purely inseparable. Take E to be a maximal separable subextension of K(a)|K; we have that E|K is nontrivial. Furthermore, E|K is a tower of Galois extensions of degree p, as K(a)|K is a tower of normal extensions of degree p. This shows that K admits an Artin-Schreier extension K(ϑ) ⊆ K(a), where ϑ is an Artin-Schreier generator. Since K(a)|K is an immediate extension of henselian fields, the same holds for K(ϑ)|K and thus K(ϑ)|K is an Artin-Schreier defect extension. Take a polynomial f ∈ K[X] such that ϑ = f (a) with deg f < p i . Since a is strongly immediate by assumption, we can apply Lemma 16 to obtain that dist (ϑ, K) = dist (f (a), K) = α + p s dist (a, K) (11) for some α ∈ vK and s < i. Take c ∈ K such that vc = α.