Perturbing Eisenstein polynomials over local fields Kevin Keating Department of Mathematics 7 University of Florida 1 0 Gainesville, FL 32611 2 USA n a J [email protected] 8 January 10, 2017 ] T N h. Abstract t a Let K be a local field whose residue field has characteristic p and let L/K be m a finite separable totally ramified extension. Let π be a uniformizer for L and L [ let f(X) be the minimum polynomial for π over K. Suppose π˜ is another L L 1 uniformizer for L such that π˜ π + rπℓ+1 (mod πℓ+2) for some ℓ 1 and v L ≡ L L L ≥ 8 r K. Let f˜(X) be the minimum polynomial for π˜L over K. In this paper we 7 giv∈eOcongruences for the coefficients of f˜(X) in terms of r and the coefficients of 9 f(X). These congruences improve and extend work of Krasner [7]. 1 0 . 1 1 Introduction 0 7 1 Let K be a field which is complete with respect to a discrete valuation v . Let be : K K v O the ring of integers of K and let be the maximal ideal of . Assume that the Xi residue field K = / of KMis Ka perfect field of characteriOstKic p. Let Ksep be a K K r separable closure ofOK aMnd let L/K be a finite totally ramified subextension of Ksep/K. a Let π be a uniformizer for L and let L f(X) = Xn c Xn−1 + +( 1)n−1c X +( 1)nc 1 n−1 n − ··· − − be the minimum polynomial of π over K. Let ℓ 1, let r , and let π˜ be another L K L ≥ ∈ O uniformizer for L such that π˜ π +rπℓ+1 (mod ℓ+2). Let L ≡ L L ML f˜(X) = Xn c˜ Xn−1 + +( 1)n−1c˜ X +( 1)nc˜ 1 n−1 n − ··· − − betheminimum polynomial ofπ˜ over K. Inthis paperwe usethetechniques developed L ˜ in[6]toobtaincongruencesforthecoefficientsc˜ off(X)intermsofr andthecoefficients i of f(X). 1 Let φ : R R be the Hasse-Herbrand function of L/K, as defined for L/K ≥0 ≥0 → instance in Chapter IV of [9]. For 1 h n set k = φ (ℓ) + h . Krasner [7, ≤ ≤ h ⌈ L/K n⌉ p.157] showed that for 1 h n we have c˜ c (mod kh). In Theorem 4.3 we prove that c˜ c (mod ≤kh′)≤for certain intehge≡rs kh′ such tMhaKt k′ k . Let h be the h ≡ h MK h h ≥ h unique integer such that 1 h n and n divides nφ (ℓ)+h. Krasner [7, p.157] gave L/K ≤ ≤ a formula for the congruence class modulo kh+1 of c˜ c . In Theorem 4.5 we give MK h − h similar formulas for up to ν +1 values of h, where ν = v (n). p Heiermann [3] gave formulas which are analogous to the results presented here. Let ¨ S be the set of Teichmuller representatives for K. Let π be a uniformizer for K K K ⊂ O and let (X) be the unique power series with coefficients in S such that π = πn (π ). F K LF L Supposeπ˜ isanotheruniformizer forLsuchthatπ˜ π +rπℓ+1 (mod ℓ+2)forsome L L ≡ L L ML ℓ 1 and r S. Let ˜ be the series with coefficients in S such that π = π˜n ˜(π˜ ). Us≥ing Theore∈m 4.6 of F[3] one can compute certain coefficients of ˜ in tKerms oLfFr aLnd F the coefficients of . F In Section 2 and we recall some facts about symmetric polynomials from [6]. The main focus is on expressing monomial symmetric polynomials in terms of elementary symmetric polynomials. In Section 3 we define the indices of inseparability of L/K and some generalizations of the function φ . In Section 4 we prove our main results. In L/K Section 5 we give some examples which illustrate how the theorems from Section 4 are applied. 2 Symmetric polynomials and cycle digraphs Let n 1, let w 1, and let µ be a partition of w. We view µ as a multiset of ≥ ≥ positive integers such that the sum Σ(µ) of the elements of µ is equal to w. The cardinality of µ is denoted by µ . For µ such that µ n we let m (X ,...,X ) be µ 1 n | | | | ≤ the monomial symmetric polynomial in n variables associated to µ. For 1 h n let ≤ ≤ e (X ,...,X ) denote the elementary symmetric polynomial of degree h in n variables. h 1 n By the fundamental theorem of symmetric polynomials there is a unique polynomial ψ Z[X ,...,X ] such that m = ψ (e ,...,e ). In this section we use a theorem of µ 1 n µ µ 1 n ∈ Kulikauskas and Remmel [8] to compute certain coefficients of ψ . µ The formula of Kulikauskas and Remmel can be expressed in terms of tilings of a certain type of digraph. We say that a directed graph Γ is a cycle digraph if it is a disjoint union of finitely many directed cycles of length 1. We denote the vertex set ≥ of Γ by V(Γ), and we define the sign of Γ to be sgn(Γ) = ( 1)w−c, where w = V(Γ) − | | and c is the number of cycles that make up Γ. Let Γ be a cycle digraph with w 1 vertices and let λ be a partition of w. A λ-tiling ≥ of Γ is a set S of subgraphs of Γ such that 1. Each γ S is a directed path of length 0. ∈ ≥ 2. The collection V(γ) : γ S forms a partition of the set V(Γ). { ∈ } 3. The multiset V(γ) : γ S is equal to λ. {| | ∈ } 2 Let µ be another partition of w. A (λ,µ)-tiling of Γ is an ordered pair (S,T), where S is a λ-tiling of Γ and T is a µ-tiling of Γ. Let Γ′ be another cycle digraph with w vertices and let (S′,T′) be a (λ,µ)-tiling of Γ′. An isomorphism from (Γ,S,T) to (Γ′,S′,T′) is an isomorphism of digraphs θ : Γ Γ′ which carries S onto S′ and T onto T′. Say that → the (λ,µ)-tilings (S,T) and (S′,T′) of Γ are isomorphic if there exists an isomorphism from (Γ,S,T) to (Γ,S′,T′). Say that (S,T) is an admissible (λ,µ)-tiling of Γ if (Γ,S,T) has no nontrivial automorphisms. Let η (Γ) denote the number of isomorphism classes λµ of admissible (λ,µ)-tilings of Γ. Let w 1 and let λ,µ be partitions of w. Set ≥ d = ( 1)|λ|+|µ| sgn(Γ)η (Γ), (2.1) λµ λµ − · Γ X where the sum is over all isomorphism classes of cycle digraphs Γ with w vertices. Since η = η we have d = d . Kulikauskas and Remmel [8, Th.1(ii)] proved the µλ λµ µλ λµ following: Theorem 2.1 Let n 1, let w 1, and let µ be a partition of w with at most n parts. ≥ ≥ Let ψ be the unique element of Z[X ,...,X ] such that m = ψ (e ,...,e ). Then µ 1 n µ µ 1 n ψ (X ,...,X ) = d X X ...X , µ 1 n λµ · λ1 λ2 λk λ X where the sum is over all partitions λ = λ ,...,λ of w such that 1 λ n for 1 k i { } ≤ ≤ 1 i k. ≤ ≤ We now recall some formulas from [6] for computing values of η (Γ). λµ Proposition 2.2 Let a,b,c,d,w be positive integers such that a = c, b = d, and let r,s 6 6 be nonnegative integers. Let Γ be a directed cycle of length w. (a) Suppose w = ra = sb + d. Let λ be the partition of w consisting of r copies of a, and let µ be the partition of w consisting of s copies of b and one copy of d. Then η (Γ) = a. λµ (b) Suppose w = ra+c = sb+d. Let λ be the partition of w consisting of r copies of a and one copy of c, and let µ be the partition of w consisting of s copies of b and one copy of d. Then η (Γ) = w. λµ Proof: Statement(a)followsfromProposition2.5of[6]ifs = 0,andfromProposition2.3 of [6] if s 1. Statement (b) follows from Proposition 2.2 of [6]. (cid:3) ≥ Using these formulas we can compute d in some cases. λµ Proposition 2.3 Let a,b,c,d,w be positive integers such that a = c and b = d. Let 6 6 r,s be nonnegative integers such that w = ra + c = sb + d and a > sb. Let λ be the 3 partition of w consisting of r copies of a and 1 copy of c, and let µ be the partition of w consisting of s copies of b and 1 copy of d. Then ( 1)r+s+w+1w if b ∤ c or sb < c, d = − λµ (( 1)r+s+w+1(w ab) if b c and sb c. − − | ≥ Proof: Let Γ be a cycle digraph which has an admissible (λ,µ)-tiling. Suppose Γ consists of a single cycle of length w. Then by Proposition 2.2(b) we have η (Γ) = w. λµ Suppose Γ has more than one cycle. Since Γ has a µ-tiling, Γ has a cycle Γ such that 1 V(Γ ) sb. Since a > sb and Γ has a λ-tiling, it follows that V(Γ ) = c = mb 1 1 | | ≤ | | for some m such that 1 m s. Hence if Γ has more than one cycle we must have ≤ ≤ b c and c sb. Let λ be the partition of c consisting of one copy of c and let µ | ≤ 1 1 be the partition of c consisting of m copies of b. Then every λ-tiling of Γ restricts to a λ -tiling of Γ , and every µ-tiling of Γ restricts to a µ -tiling of Γ . It follows from 1 1 1 1 Proposition 2.2(a) that η (Γ ) = b. λ1µ1 1 Let Γ be another cycle of Γ. Since Γ has a λ-tiling, V(Γ ) a > sb. Hence 2 2 | | ≥ every µ-tiling of Γ restricts to a tiling of Γ which includes a path δ with V(δ) = d. 2 | | Since µ has only one part equal to d, it follows that Γ = Γ Γ . Therefore we have 1 2 ∪ V(Γ ) = ra = (s m)b+ d. Let λ be the partition of ra consisting of r copies of a 2 2 | | − and let µ be the partition of (s m)b+d = ra consisting of s m copies of b and 1 2 − − copy of d. Then every λ-tiling of Γ restricts to a λ -tiling of Γ , and every µ-tiling of 2 2 Γ restricts to a µ -tiling of Γ . It follows from Proposition 2.2(a) that η (Γ ) = a. 2 2 λ2µ2 2 Hence η (Γ) = η (Γ ) η (Γ ) = ba. λµ λ1µ1 1 · λ2µ2 2 Suppose b ∤ c or c > sb. Then it follows from the above that the only cycle digraph which has a (λ,µ)-tiling consists of a single cycle of length w. Hence by (2.1) we get d = ( 1)(r+1)+(s+1) ( 1)w−1w. λµ − · − Suppose b c and sb c. Then c = mb with 1 m s. Hence there are two cycle | ≥ ≤ ≤ digraphs which have a (λ,µ)-tiling: a single cycle of length w, and the union of two cycles with lengths c = mb and ra = (s m)b+d. Therefore by (2.1) we get − d = ( 1)(r+1)+(s+1)(( 1)w−1w +( 1)w−2ab). λµ − − − Hence the formula for d given in the theorem holds in both cases. (cid:3) λµ We recall some results from[6] regarding the p-adic properties of the coefficients d . λµ Let w 1 and let λ be a partition of w. For k 1 let k λ be the partition of kw ≥ ≥ ∗ which is the multiset sum of k copies of λ, and let k λ be the partition of kw obtained · by multiplying the parts of λ by k. Proposition 2.4 Let t j 0, let w′ 1, and set w = w′pt. Let λ′ be a partition of ≥ ≥ ≥ w′ and set λ = pt λ′. Let µ be a partition of w such that there does not exist a partition · µ′ with µ = pj+1 µ′. Then pt−j divides d . λµ ∗ 4 (cid:3) Proof: This is proved in Corollary 3.4 of [6]. Proposition 2.5 Let w′ 1, j 1, and t 0. Let λ′, µ′ be partitions of w′ such that ≥ ≥ ≥ the parts of λ′ are all divisible by pt. Set w = w′pj, so that λ = pj λ′ and µ = pj µ′ · ∗ are partitions of w. Then dλµ dλ′µ′ (mod pt+1). ≡ (cid:3) Proof: This is proved in Proposition 3.5 of [6]. 3 Indices of inseparability Let L/K be a totally ramified extension of degree n = upν, with p ∤ u. Let π be a L uniformizer for L whose minimum polynomial over K is f(X) = Xn c Xn−1 + +( 1)n−1c X +( 1)nc . 1 n−1 n − ··· − − For k Z define v (k) = min v (k),ν . For 0 j ν set p p ∈ { } ≤ ≤ iπL = min nv (c ) h : 1 h n, v (h) j (3.1) j { K h − ≤ ≤ p ≤ } = min v (c πn−h) : 1 h n, v (h) j n. { L h L ≤ ≤ p ≤ }− Then iπL is either a nonnegative integer or ; if char(K) = p then iπL must be finite, j ∞ j since L/K is separable. Let e = v (p) denote the absolute ramification index of L. We L L define the jth index of inseparability of L/K to be i = min iπL +(j′ j)e : j j′ ν . (3.2) j { j′ − L ≤ ≤ } By Proposition 3.12 and Theorem 7.1 of [3], i does not depend on the choice of π . j L Furthermore, our definition of i agrees with Definition 7.3 in [3]; for the characteristic-p j case see also [1, pp.232–233] and [2, 2]. Write i = A n b with 1 b n. j j j j § − ≤ ≤ Remark 3.1 If iπL is finite we can write iπL = a n b with a 1 (see Section 4 of j j j − j j ≥ [6]). Thus if ij = iπj′L +(j′ −j)eL then Aj = aj′ +(j′ −j)eK. The following facts are easy consequences of the definitions: 1. 0 = i < i i i < . ν ν−1 1 0 ≤ ··· ≤ ≤ ∞ 2. If char(K) = p then i = iπL. j j 3. Let m = v (i ). If m j then i = i = iπL = iπL. If m > j then char(K) = 0 p j ≤ j m j m and i = iπL +(m j)e . j m − L ˜ Following [3, (4.4)], for 0 j ν we define functions φ : [0, ) [0, ) by j φ˜ (x) = i + pjx. The general≤ized H≤asse-Herbrand functions φ : [0,∞) → [0, ∞) are j j j ∞ → ∞ then defined by ˜ φ (x) = min φ (x) : 0 j j . (3.3) j { j0 ≤ 0 ≤ } Hence we have φj(x) φj′(x) for 0 j′ j. Let φL/K : [0, ) [0, ) be the usual ≤ ≤ ≤ ∞ → ∞ Hasse-Herbrand function. Then by Corollary 6.11 of [3] we have φ (x) = nφ (x). ν L/K For a partition λ = λ ,...,λ whose parts satisfy 1 λ n define c = 1 k i λ { } ≤ ≤ c c ...c . The following is proved in Proposition 4.2 of [6]. λ1 λ2 λk 5 Proposition 3.2 Let w 1 and let λ = λ ,...,λ be a partition of w whose parts 1 k ≥ { } satisfy 1 λ n. Choose q to minimize v (λ ) and set t = v (λ ). Then v (c ) i p q p q L λ ≤ ≤ ≥ iπL +w. If v (c ) = iπL +w and iπL < then λ = b and λ = b = n for all i = q. t L λ t t ∞ q t i ν 6 4 Perturbing π L Inthissectionweproveourmaintheorems. WebeginbyapplyingtheresultsofSection2 to the totally ramified extension L/K. Write [L : K] = n = upν with p ∤ u. Let π , π˜ L L be uniformizers for L, with minimum polynomials over K given by f(X) = Xn c Xn−1 + +( 1)n−1c X +( 1)nc 1 n−1 n − ··· − − f˜(X) = Xn c˜ Xn−1 + +( 1)n−1c˜ X +( 1)nc˜ . 1 n−1 n − ··· − − Let 1 h n and set j = v (h). Define a function ρ : N N by p h ≤ ≤ → φ (ℓ)+h j ρ (ℓ) = . h n (cid:24) (cid:25) Let ℓ 1. We say f˜ f if c˜ c (mod ρh(ℓ)) for 1 h n. Thus is an ≥ ∼ℓ h ≡ h MK ≤ ≤ ∼ℓ equivalence relation on the set of minimum polynomials over K for uniformizers of L. Let σ ,...,σ be the K-embeddings of L into Ksep. For each partition µ with at 1 n most n parts define M : L K by µ → M (α) = m (σ (α),...,σ (α)). µ µ 1 n For 1 h n define E : L K by h ≤ ≤ → E (α) = e (σ (α),...,σ (α)). h h 1 n Then c = E (π ) and c˜ = E (π˜ ). h h L h h L Proposition 4.1 Let φ(X) = r X + r X2 + be a power series with coefficients in 1 2 ··· such that π˜ = φ(π ). Then for 1 h n we have K L L O ≤ ≤ E (π˜ ) = r r ...r M (π ), h L µ1 µ2 µh µ L µ X where the sum ranges over all partitions µ = µ ,...,µ with h parts. 1 h { } (cid:3) Proof: This is a special case of Proposition 4.4 in [6]. Proposition 4.2 Let n 1, let w 1, and let µ be a partition of w with at most n ≥ ≥ parts. Then M (π ) = d c , µ L λµ λ λ X where the sum is over all partitions λ = λ ,...,λ of w such that 1 λ n for 1 k i { } ≤ ≤ 1 i k. ≤ ≤ 6 Proof: This follows from Theorem 2.1 by setting X = E (π ) = c . (cid:3) i i L i Let ℓ 1. Our first main result gives congruences between the coefficients of f(X) and the c≥oefficients of f˜(X) under the assumption π˜ π (mod ℓ+1). L ≡ L ML Theorem 4.3 Let π , π˜ be uniformizers for L and let f(X), f˜(X) be the minimum L L polynomials for π , π˜ over K. Suppose there are ℓ 1 and σ Aut (L) such that L L K σ(π˜ ) π (mod ℓ+1). Then f˜ f. ≥ ∈ L ≡ L ML ∼ℓ Proof: We first show that the theorem holds in the case where π˜ = π +rπℓ+1, with L L L r . Let 1 h n and set j = v (h). For 0 s h let µ be the partition of ∈ OK ≤ ≤ p ≤ ≤ s ℓs+h consisting of h s copies of 1 and s copies of ℓ+1. Then by Proposition 4.1 we − have h h c˜ = E (π˜ ) = M (π )rs = c + M (π )rs. (4.1) h h L µ L h µ L s s s=0 s=1 X X To prove that c˜ c (mod ρh(ℓ)) it’s enough to show that v (M (π )) ρ (ℓ) for h ≡ h MK K µs L ≥ h 1 s h. Therefore by Proposition 4.2 it suffices to show v (d c ) φ (ℓ)+h for L λµ λ j ≤ ≤ s ≥ all 1 s h and all partitions λ of ℓs+h whose parts are at most n. ≤ ≤ Let 1 s h, set j = v (h), and set m = min j,v (s) . Then m j and s pm. p p ≤ ≤ { } ≤ ≥ Let λ = λ ,...,λ be a partition of ℓs+h such that 1 λ n for 1 i k. Choose 1 k i { } ≤ ≤ ≤ ≤ q tominimizev (λ )andsett = v (λ ). ByProposition3.2wehavev (c ) iπL+ℓs+h. p q p q L λ ≥ t Suppose m < t. Then m < ν, so we have pm+1 ∤ gcd(h s,s). Hence by Proposition 2.4 − we get v (d ) t m. Thus p λµ s ≥ − v (d c ) = v (d )+v (c ) L λµ λ L λµ L λ s s (t m)v (p)+iπL +ℓs+h ≥ − L t i +ℓpm +h. m ≥ Suppose m t. Then ≥ v (d c ) v (c ) L λµ λ L λ s ≥ iπL +ℓs+h ≥ t i +ℓpm +h t ≥ i +ℓpm +h. m ≥ Inbothcaseswegetv (d c ) φ˜ (ℓ)+h φ (ℓ)+h,andhencec˜ c (mod ρh(ℓ)). L λµs λ ≥ m ≥ j h ≡ h MK ˜ Since this holds for 1 h n we get f f. ℓ ≤ ≤ ∼ ˜ We now prove the general case. Since f is the minimum polynomial of σ(π˜ ) over L K we may assume without loss of generality that π˜ π (mod ℓ+1). By repeated L ≡ L ML (0) (1) (2) application of the special case above we get a sequence π = π ,π ,π ,... of uni- L L L L formizers for L with minimum polynomials f(0) = f,f(1),f(2),... such that for all i 0 we have π(i) π˜ (mod ℓ+i+1) and f(i+1) f(i). It follows that f(i+1) f(i), a≥nd L ≡ L ML ∼ℓ+i ∼ℓ hence that f(i) f for all i 0. Since the sequence (f(i)) converges coefficientwise to ℓ f˜it follows that∼f˜ f. ≥ (cid:3) ℓ ∼ 7 Remark 4.4 It follows from Theorem 4.3 that if σ(π˜ ) π (mod ℓ+1) for some L ≡ L ML σ Aut (L) then c˜ c (mod ρh(ℓ)) for 1 h n. Define functions κ : N N ∈ K h ≡ h MK ≤ ≤ h → by φ (ℓ)+h ν κ (ℓ) = . h n (cid:24) (cid:25) Krasner [7, p.157] showed that c˜ c (mod κh(ℓ)). Since κ (ℓ) ρ (ℓ) Krasner’s h ≡ h MK h ≤ h congruences are in general weaker than the congruences that follow from Theorem 4.3. However, if ℓ is greater than or equal to the largest lower ramification break of L/K then φ (ℓ) = φ (ℓ) for 0 j ν. Therefore Theorem 4.3 does not improve on [7] in j ν ≤ ≤ these cases. For certain values of h we get a more refined version of the congruences that follow from Theorem 4.3. Theorem 4.5 Let L/K be a finite totally ramified extension of degree n = upν. For 0 m ν write the mth index of inseparability of L/K in the form i = A n b m m m ≤ ≤ − with 1 b n. Let π , π˜ be uniformizers for L such that there are ℓ 1, r , m L L K ≤ ≤ ≥ ∈ O and σ Aut (L) with σ(π˜ ) π + rπℓ+1 (mod ℓ+2). Let 0 j ν satisfy ∈ K L ≡ L L ML ≤ ≤ v (φ (ℓ)) = j, and let h be the unique integer such that 1 h n and n divides p j ≤ ≤ φ (ℓ)+h. Set k = (φ (ℓ)+h)/n and h = h/pj. Then j j 0 c˜ c + g ck−Amc rpm (mod k+1), h ≡ h m n bm MK mX∈Sj where ˜ S = m : 0 m j, φ (ℓ) = φ (ℓ) j j m { ≤ ≤ } ( 1)k+ℓ+Am(h pj−m +ℓ upν−m) if b < h 0 m − − gm = ( 1)k+ℓ+Am(h0pj−m +ℓ) if h bm < n − ≤ ( 1)k+ℓ+Amupν−m if b = n. m −  Proof: We first provethat the theorem holds for πˆ = π +rπℓ+1. Let L L L fˆ(X) = Xn cˆ Xn−1 + +( 1)n−1cˆ X +( 1)ncˆ 1 n−1 n − ··· − − be the minimum polynomial for πˆ over K. Let 1 s h and let λ be a partition of L ≤ ≤ ℓs+hwhose parts areat most n. Choose q to minimize v (λ ) and set t = v (λ ). Recall p q p q that µ is the partition of ℓs+h consisting of h s copies of 1 and s copies of ℓ+1. Since s − v (h) = v (φ (ℓ)) = j it follows from the proof of Theorem 4.3 that v (d c ) k. p p j K λµ λ s ≥ Suppose v (d c ) = k. Then the inequalities in the proof of Theorem 4.3 must be K λµ λ s equalities. Hence there is 0 m j such that s = pm, v (c ) = iπL + ℓpm + h, and ˜ ≤ ≤ L λ t φ (ℓ) = φ (ℓ). In particular, we have m S . j m j ∈ 8 Let w = ℓpm + h and let κ be the partition of w consisting of k A copies m m m m − of n and 1 copy of b . By Proposition 3.2 we see that λ has at most one element not m equal to n. Since λ is a partition of w , and m w = φ (ℓ) i +h = (k A )n+b , m j m m m − − it follows that λ = κ . Hence c = c = ck−Amc and v (b ) = v (λ ) = t. Using m λ κm n bm p m p q equation (4.1) and Proposition 4.2 we get cˆ c + d ck−Amc rpm (mod k+1). (4.2) h ≡ h κmµpm n bm MK mX∈Sj Let m S . Since j ∈ j = v (φ (ℓ)) = v (φ˜ (ℓ)) = v (i +ℓpm) p j p m p m and m j we get m v (i ) = v (b ). Hence b′ = b /pm is an integer. Let κ′ be ≤ ≤ p m p m m m m the partition of w′ = (k A )upν−m +b′ = h pj−m +ℓ m − m m 0 consisting of k A copies of upν−m and 1 copy of b′ . Let µ′ be the partition − m m pm of w′ consisting of h pj−m 1 copies of 1 and 1 copy of ℓ + 1. Since h n we m 0 − ≤ have upν−m > h0pj−m − 1. Hence if b′m 6= upν−m then we can compute dκ′mµ′pm using Proposition 2.3. Suppose b < h. Then h pj−m 1 b′ , so by Proposition 2.3 we get m 0 − ≥ m dκ′mµ′pm = (−1)k+ℓ+Am(h0pj−m +ℓ−upν−m). Suppose h b < n. Then h pj−m 1 < b′ , so by Proposition 2.3 we get ≤ m 0 − m dκ′mµ′pm = (−1)k+ℓ+Am(h0pj−m +ℓ). Suppose b = n, so that b′ = upν−m. Since upν−m > h pj−m 1, the only cycle m m 0 − digraph which admits a (κ′ ,µ′ )-tiling consists of a single cycle Γ of length w′ . By m pm m Proposition 2.2(a) we get ηκ′mµ′pm(Γ) = upν−m. It then follows from (2.1) that dκ′mµ′pm = (−1)k+ℓ+Amupν−m. Hence in all three cases we have dκ′mµ′pm = gm. Since m t ν it follows from (3.2) and (3.1) that ≤ ≤ i iπL +(t m)e m ≤ t − L nA b nv (c ) b +(t m)e m − m ≤ K bm − m − L A v (c )+(t m)e m ≤ K bm − K k +1 k A +v (c )+(t m+1)e . (4.3) ≤ − m K bm − K 9 Since pt b we have pt−m b′ . Therefore by Proposition 2.5 we get | m | m dκmµpm ≡ dκ′mµ′pm (mod pt−m+1). Using (4.3) we see that dκmµpmckn−Amcbm ≡ dκ′mµ′pmckn−Amcbm (mod MkK+1) g ck−Amc (mod k+1). ≡ m n bm MK Therefore the theorem holds when π˜ = πˆ . L L We now prove the theorem in the general case. We may assume that π˜ π +rπℓ+1 (mod ℓ+2). L ≡ L L ML Itfollowsthatπ˜ πˆ (mod ℓ+2), sobyTheorem4.3wegetc˜ cˆ (mod ρh(ℓ+1)). L ≡ L ML h ≡ h MK Since (φ (ℓ)+h)/n = k and φ (ℓ+1) > φ (ℓ) this implies c˜ cˆ (mod k+1). Hence j j j h ≡ h MK the theorem holds for π˜ . (cid:3) L Remark 4.6 Suppose vp(φj(ℓ)) = j′ j. Then φj(ℓ) = φj′(ℓ). In particular, φν(ℓ) = ≤ φj′(ℓ) with j′ = vp(φν(ℓ)). Hence if 1 h n and n divides φν(ℓ)+h then Theorem 4.5 ≤ ≤ gives a congruence for c˜ modulo k+1, where k = (φ (ℓ)+h)/n. This is the congru- h MK ν ence obtained by Krasner [7, p.157]. If ℓ is greater than or equal to the largest lower ramification break of L/K then φ (ℓ) = φ (ℓ) for 0 j ν. Therefore Theorem 4.5 j ν ≤ ≤ does not extend [7] in these cases. 5 Some examples In this section we give two examples related to the theorems proved in Section 4. We first apply these theorems to a 3-adic extension of degree 9: Example 5.1 Let K be a finite extension of the 3-adic field Q such that v (3) 2. 3 K ≥ Let f(X) = X9 c X8 + +c X c 1 8 9 − ··· − be an Eisenstein polynomial over K such that v (c ) = v (c ) = 2, v (c ) 2 for K 2 K 6 K h ≥ h 1,3 , and v (c ) 3 for h 4,5,7,8 . Let π be a root of f(X). Then K h L ∈ { } ≥ ∈ { } L = K(π ) is a totally ramified extension of K of degree 9, so we have u = 1, ν = 2. It L follows from our assumptions about the valuations of the coefficients of f(X) that the indices of inseparability of L/K are i = 16, i = 12, and i = 0. Therefore A = 2, 0 1 2 0 ˜ A = 2, A = 1, and b = 2, b = 6, b = 9. We get the following values for φ (ℓ) and 1 2 0 1 2 j φ (ℓ): j ˜ ˜ ˜ ℓ φ (ℓ) φ (ℓ) φ (ℓ) φ (ℓ) φ (ℓ) φ (ℓ) 0 1 2 0 1 2 1 17 15 9 17 15 9 2 18 18 18 18 18 18 3 19 21 27 19 19 19 10