Extensions of local fields and elementary symmetric polynomials
aa r X i v : . [ m a t h . N T ] A ug Extensions of local fields and elementary symmetricpolynomials
Kevin KeatingDepartment of MathematicsUniversity of FloridaGainesville, FL 32611USA [email protected]
October 8, 2018
Abstract
Let K be a local field whose residue field has characteristic p and let L/K be afinite separable totally ramified extension of degree n = up ν . Let σ , . . . , σ n denotethe K -embeddings of L into a separable closure K sep of K . For 1 ≤ h ≤ n let e h ( X , . . . , X n ) denote the h th elementary symmetric polynomial in n variables,and for α ∈ L set E h ( α ) = e h ( σ ( α ) , . . . , σ n ( α )). Set j = min { v p ( h ) , ν } . Weshow that for r ∈ Z we have E h ( M rL ) ⊂ M ⌈ ( i j + hr ) /n ⌉ K , where i j is the j th index ofinseparability of L/K . In certain cases we also show that E h ( M rL ) is not containedin any higher power of M K . Let K be a field which is complete with respect to a discrete valuation v K . Let O K bethe ring of integers of K and let M K be the maximal ideal of O K . Assume that theresidue field K = O K / M K of K is a perfect field of characteristic p . Let K sep be aseparable closure of K , and let L/K be a finite totally ramified subextension of K sep /K of degree n = up ν , with p ∤ u . Let σ , . . . , σ n denote the K -embeddings of L into K sep .For 1 ≤ h ≤ n let e h ( X , . . . , X n ) denote the h th elementary symmetric polynomial in n variables, and define E h : L → K by setting E h ( α ) = e h ( σ ( α ) , . . . , σ n ( α )) for α ∈ L .We are interested in the relation between v L ( α ) and v K ( E h ( α )). In particular, for r ∈ Z we would like to compute the value of g h ( r ) = min { v K ( E h ( α )) : α ∈ M rL } . The following proposition shows that g h ( r ) is a well-defined integer:1 roposition 1.1 Let
L/K be a totally ramified extension of degree n . Let r ∈ Z andlet h satisfy ≤ h ≤ n . Then E h ( M rL ) ⊂ M ⌈ hr/n ⌉ K and E h ( M rL ) = { } .Proof: For the first claim we observe that if α ∈ M rL then v L ( E h ( α )) ≥ hr , andhence v K ( E h ( α )) ≥ hr/n . To prove the second claim let A = L ⊗ K K sep and define˜ E h : A → K sep as follows. For β ∈ A define T β : A → A by T β ( x ) = βx for x ∈ A . Then T β is a K sep -linear map. Letdet( X · id A − T β ) = X n − c X n − + · · · + ( − n − c n − X + ( − n c n be the characteristic polynomial of T β , and set ˜ E h ( β ) = c h . Since L/K is separable wehave an isomorphism of K sep -algebras A ∼ = ( K sep ) n . It follows that ˜ E h is onto. Thereis an embedding of K -algebras i : L → A defined by i ( α ) = α ⊗ α ∈ L . It followsfrom the definitions that ˜ E h ◦ i = E h .Let { v , . . . , v n } be a basis for L over K . Then { v ⊗ , . . . , v n ⊗ } is a basis for A over K sep . For x i ∈ K sep define F ( x , . . . , x n ) = ˜ E h ( v ⊗ x + · · · + v n ⊗ x n ) . Then F is a degree- h form on ( K sep ) n . Furthermore, since ˜ E h is onto, F in nontrivial.Since K is an infinite field there are d i ∈ K such that F ( d , . . . , d n ) = 0. Set α = d v + · · · + d n v n . Then α ∈ L and E h ( α ) = ˜ E h ( α ⊗
1) = F ( d , . . . , d n ) = 0 . Let π K be a uniformizer for K . Then for t sufficiently large we have π tK α ∈ M rL and E h ( π tK α ) = π htK E h ( α ) = 0. Hence E h ( M rL ) = { } . (cid:3) Since
L/K is totally ramified we have v K ( E n ( α )) = v K (N L/K ( α )) = v L ( α ) , and hence g n ( r ) = r for r ∈ Z . The map E = Tr L/K is also well-understood, at leastwhen
L/K is a Galois extension of degree p (see [7, V §
3, Lemma 4] or [1, III, Prop. 1.4]).
Proposition 1.2
Let
L/K be a totally ramified extension of degree n and let M dL bethe different of L/K . Then for every r ∈ Z we have E ( M rL ) = M ⌊ ( d + r ) /n ⌋ K . Therefore g ( r ) = ⌊ ( d + r ) /n ⌋ .Proof: Since E ( M rL ) is a nonzero fractional ideal of K we have E ( M rL ) = M sK forsome z ∈ Z . By Proposition 7 in [7, III §
3] we have M d + rL ⊂ O L · M sK = M nsL M d + rL
6⊂ O L · M s +1 K = M n ( s +1) L . It follows that d + r ≥ ns and d + r < n ( s + 1), and hence that s = ⌊ ( d + r ) /n ⌋ . (cid:3)
2n this paper we determine a lower bound for g h ( r ) which depends on the indices ofinseparability of L/K . When h = p j with 0 ≤ j ≤ ν and K is large enough we showthat g h ( r ) is equal to this lower bound. This leads to a formula for g p j ( r ) which can beexpressed in terms of a generalization of the different of L/K (see Remark 5.4).In sections 2 and 3 we prove some preliminary results involving symmetric polyno-mials. The main focus is on expressing monomial symmetric polynomials in terms ofelementary symmetric polynomials. In section 4 we prove our lower bound for g h ( r ). Insection 5 we show that g h ( r ) is equal to this lower bound in some special cases. Let n ≥
1, let w ≥
1, and let λ be a partition of w . We view λ as a multiset of positiveintegers such that the sum Σ( λ ) of the elements of λ is equal to w . The cardinality of λ is denoted by | λ | . For k ≥ k ∗ λ be the partition of kw which is the multiset sumof k copies of λ , and we let k · λ be the partition of kw obtained by multiplying the partsof λ by k . If | λ | ≤ n let m λ ( X , . . . , X n ) be the monomial symmetric polynomial in n variables associated to λ . For 1 ≤ h ≤ n let e h ( X , . . . , X n ) denote the h th elementarysymmetric polynomial in n variables.Let r ≥ φ ( X ) = a r X r + a r +1 X r +1 + · · · be a power series with genericcoefficients a i . For a partition µ = { µ , . . . , µ h } whose parts satisfy µ i ≥ r set a µ = a µ a µ . . . a µ h . Then for 1 ≤ h ≤ n we have e h ( φ ( X ) , . . . , φ ( X n )) = X µ a µ m µ ( X , . . . , X n ) , (2.1)where the sum ranges over all partitions µ with h parts, all of which are ≥ r . By thefundamental theorem of symmetric polynomials there is ψ µ ∈ Z [ X , . . . , X n ] such that m µ = ψ µ ( e , . . . , e n ). In this section we use a theorem of Kulikauskas and Remmel [6]to compute certain coefficients of the polynomials ψ µ .The formula of Kulikauskas and Remmel can be expressed in terms of tilings of acertain type of digraph. We say that a directed graph Γ is a cycle digraph if it is adisjoint union of finitely many directed cycles of length ≥
1. We denote the vertex setof Γ by V (Γ), and we define the sign of Γ to be sgn(Γ) = ( − 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 ≥ λ be a partition of w . A λ -tilingof Γ is a set S of subgraphs of Γ such that1. Each γ ∈ S is a directed path of length ≥ { V ( γ ) : γ ∈ S } forms a partition of the set V (Γ).3. The multiset {| V ( γ ) | : γ ∈ S } is equal to λ .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 with3 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 ( S, T ) is an admissible ( λ , µ )-tiling of Γ if (Γ , S, T ) has no nontrivialautomorphisms. Say that the ( λ , µ )-tilings ( S, T ) and ( S ′ , T ′ ) of Γ are isomorphic ifthere exists an isomorphism from (Γ , S, T ) to (Γ , S ′ , T ′ ). Let η λµ (Γ) denote the numberof isomorphism classes of admissible ( λ , µ )-tilings of Γ.Let w ≥ λ , µ be partitions of w . Set d λµ = ( − | λ | + | µ | · X Γ sgn(Γ) η λµ (Γ) , (2.2)where the sum is over all isomorphism classes of cycle digraphs Γ with w vertices. Since η µλ = η λµ we have d µλ = d λµ . Kulikauskas and Remmel [6, Th. 1(ii)] proved thefollowing: Theorem 2.1
Let n ≥ , let w ≥ , and let µ be a partition of w with at most n parts.Let ψ µ be the unique element of Z [ X , . . . , X n ] such that m µ = ψ µ ( e , . . . , e n ) . Then ψ µ ( X , . . . , X n ) = X λ d λµ · X λ X λ . . . X λ k , where the sum is over all partitions λ = { λ , . . . , λ k } of w such that λ i ≤ n for ≤ i ≤ k . The remainder of this section is devoted to computing the values of η λµ (Γ) and d λµ in some special cases. Proposition 2.2
Let w ≥ , let λ , µ be partitions of w , and let Γ be a directed cycleof length w . Assume that Γ has a λ -tiling S which is unique up to isomorphism, andthat Aut(Γ , S ) is trivial. Similarly, assume that Γ has a µ -tiling T which is unique upto isomorphism, and that Aut(Γ , T ) is trivial. Then η λµ (Γ) = w .Proof: For 0 ≤ i < w let S i be the rotation of S by i steps. Then the isomorphismclasses of ( λ , µ )-tilings of Γ are represented by ( S i , T ) for 0 ≤ i < w . Since Aut(Γ , T ) istrivial, all these tilings are admissible. (cid:3) Proposition 2.3
Let a, b, c, ℓ, m, w be positive integers such that ℓa = mb + c = w and b = c . Let λ be the partition of w consisting of ℓ copies of a , let µ be the partition of w consisting of m copies of b and 1 copy of c , and let Γ be a directed cycle of length w .Then η λµ (Γ) = a .Proof: The cycle digraph Γ has a λ -tiling S which us unique up to isomorphism, anda µ -tiling T which is unique up to isomorphism. For 0 ≤ i < a let S i be the rotationof S by i steps. Then the isomorphism classes of ( λ , µ )-tilings of Γ are represented by( S i , T ) for 0 ≤ i < a . Since Aut(Γ , T ) is trivial, all these tilings are admissible. (cid:3) roposition 2.4 Let b, c, m, w be positive integers such that mb + c = w and b = c .Let λ be the partition of w consisting of 1 copy of w and let µ be the partition of w consisting of m copies of b and 1 copy of c . Then d λµ = ( − w + m +1 w .Proof: If the cycle digraph Γ has a λ -tiling then Γ consists of a single cycle of length w . Hence by (2.2) we get d λµ = ( − w + m +1 η λµ (Γ). It follows from Proposition 2.3 that η λµ (Γ) = w . Therefore d λµ = ( − w + m +1 w . (cid:3) Proposition 2.5
Let a, b, ℓ, m, w be positive integers such that ℓa = mb = w . Let λ bethe partition of w consisting of ℓ copies of a , let µ be the partition of w consisting of m copies of b , and let Γ be a directed cycle of length w .(a) The number of isomorphism classes of ( λ , µ ) -tilings of Γ is gcd( a, b ) .(b) Let ( S, T ) be a ( λ , µ ) -tiling of Γ . Then the order of Aut(Γ , S, T ) is gcd( ℓ, m ) .Proof: (a) Identify V (Γ) with Z /w Z and consider the translation action of b Z /w Z on( Z /w Z ) / ( a Z /w Z ). The isomorphism classes of ( λ , µ )-tilings of Γ correspond to theorbits of this action, and these orbits correspond to cosets of a Z + b Z = gcd( a, b ) · Z in Z .(b) The automorphisms of (Γ , S, T ) are rotations of Γ by m steps, where m is a multipleof both a and b . Hence the number of automorphisms is w/ lcm( a, b ), which is easilyseen to be equal to gcd( ℓ, m ). (cid:3) The following proposition generalizes the second part of [6, Th. 6].
Proposition 2.6
Let a, b, ℓ, m, w be positive integers such that ℓa = mb = w . Let λ bethe partition of w consisting of ℓ copies of a and let µ be the partition of w consistingof m copies of b . Set u = gcd( a, b ) and v = gcd( ℓ, m ) . Then d λµ = ( − w − v + ℓ + m (cid:18) uv (cid:19) .In particular, if u < v then d λµ = 0 .Proof: Set i = a/u and j = b/u . Then m = vi and ℓ = vj . Let Γ be a cycle digraphwhich has an admissible ( λ , µ )-tiling, and let Γ be one of the cycles which makes up Γ.Then the length of Γ is divisible by lcm( a, b ) = uij . Suppose Γ has length k · uij . Let λ be the partition of kuij consisting of kj copies of a = ui , and let µ be the partitionof kuij consisting of ki copies of b = uj . Then by Proposition 2.5(b) every ( λ , µ )-tiling of Γ has automorphism group of order gcd( ki, kj ) = k . Since Γ has an admissible( λ , µ )-tiling we must have k = 1. Therefore Γ consists of v cycles, each of length uij .By Proposition 2.5(a) the number of isomorphism classes of ( λ , µ )-tilings of a uij -cycle Γ is gcd( a, b ) = u . An admissible ( λ , µ )-tiling of Γ consists of v nonisomorphic( λ , µ )-tilings of uij -cycles. Hence the number of isomorphism classes of admissible( λ , µ )-tilings of Γ is η λµ (Γ) = (cid:18) uv (cid:19) . Hence by (2.2) we get d λµ = ( − w − v + ℓ + m (cid:18) uv (cid:19) . (cid:3) Some subrings of Z [ X , . . . , X n ] Let n ≥
1. In some cases we can get information about the coefficients d λµ whichappear in the formula for ψ µ given in Theorem 2.1 by working directly with the ring Z [ X , . . . , X n ]. In this section we define a family of subrings of Z [ X , . . . , X n ]. We thenstudy the p -adic properties of the coefficients d λµ by showing that for certain partitions µ the polynomial ψ µ is an element of one of these subrings.For k ≥ R k of Z [ X , . . . , X n ] by R k = Z [ X p k , . . . , X p k n ] + p Z [ X p k − , . . . , X p k − n ] + · · · + p k Z [ X , . . . , X n ] . We can characterize R k as the set of F ∈ Z [ X , . . . , X n ] such that for 1 ≤ i ≤ k thereexists F i ∈ Z [ X , . . . , X n ] such that F ( X , . . . , X n ) ≡ F i ( X p i , . . . , X p i n ) (mod p k +1 − i ) . (3.1) Lemma 3.1
Let k, ℓ ≥ and let F ∈ R k . Then p ℓ F ∈ R k + ℓ and F p ℓ ∈ R k + ℓ .Proof: The first claim is clear. To prove the second claim with ℓ = 1 we note that for1 ≤ i ≤ k it follows from (3.1) that F ( X , . . . , X n ) p ≡ F i ( X p i , . . . , X p i n ) p (mod p k +2 − i ) . In particular, the case i = k gives F ( X , . . . , X n ) p ≡ F k ( X p k , . . . , X p k n ) p (mod p ) ≡ F k ( X p k +1 , . . . , X p k +1 n ) (mod p ) . It follows that F p ∈ R k +1 . By induction we get F p ℓ ∈ R k + ℓ for ℓ ≥ (cid:3) Lemma 3.2
Let k, ℓ ≥ and let F ∈ R k . Then for any ψ , . . . , ψ n ∈ R ℓ we have F ( ψ , . . . , ψ n ) ∈ R k + ℓ .Proof: Since F ∈ R k we have F ( X , . . . , X n ) = k X i =0 p k − i φ i ( X p i , . . . , X p i n )for some φ i ∈ Z [ X , . . . , X n ]. Since ψ j ∈ R ℓ , by Lemma 3.1 we get ψ p i j ∈ R i + ℓ . Since R i + ℓ is a subring of Z [ X , . . . , X n ] it follows that φ i ( ψ p i , . . . , ψ p i n ) ∈ R i + ℓ . By Lemma 3.1we get p k − i φ i ( ψ p i , . . . , ψ p i n ) ∈ R k + ℓ . We conclude that F ( ψ , . . . , ψ n ) ∈ R k + ℓ . (cid:3) Proposition 3.3
Let w ≥ and let λ be a partition of w with at most n parts. For j ≥ let λ j = p j · λ . Then ψ λ j ∈ R j . roof: We use induction on j . The case j = 0 is trivial. Let j ≥ ψ λ j ∈ R j . Since λ j +1 = p · λ j we get m λ j +1 ( X , . . . , X n ) = m λ j ( X p , . . . , X pn )= ψ λ j ( e ( X p , . . . , X pn ) , . . . , e n ( X p , . . . , X pn )) . Since X pj ∈ R it follows from Lemma 3.2 that e i ( X p , . . . , X pn ) = θ i ( e , . . . , e n )for some θ i ∈ R . Therefore ψ λ j +1 ( X , . . . , X n ) = ψ λ j ( θ ( X , . . . , X n ) , . . . , θ n ( X , . . . , X n )) . By Lemma 3.2 we get ψ λ j +1 ∈ R j +1 . (cid:3) Corollary 3.4
Let t ≥ j ≥ , let w ′ ≥ , and set w = w ′ p t . Let λ ′ be a partition of w ′ and set λ = p t · λ ′ . Let µ be a partition of w such that there does not exist a partition µ ′ with µ = p j +1 ∗ µ ′ . Then p t − j divides d λµ . This holds in particular if p j +1 ∤ | µ | .Proof: Since d λµ does not depend on n we may assume without loss of generality that n ≥ w . It follows from this assumption that | λ | ≤ n , so by Proposition 3.3 we have ψ λ ∈ R t . Since w ≤ n the parts of µ = { µ , . . . , µ h } satisfy µ i ≤ n for 1 ≤ i ≤ h .Therefore the formula for ψ λ given by Theorem 2.1 includes the term d µλ X µ X µ . . . X µ h .The assumption on µ implies that X µ X µ . . . X µ h is not a p j +1 power. Since ψ λ ∈ R t this implies that p t − j divides d µλ . Since d λµ = d µλ we get p t − j | d λµ . (cid:3) Proposition 3.5
Let w ′ ≥ , j ≥ , and t ≥ . Let λ ′ , µ ′ be partitions of w ′ such thatthe parts of λ ′ are all divisible by p t . Set w = w ′ p j , so that λ = p j · λ ′ and µ = p j ∗ µ ′ are partitions of w . Then d λµ ≡ d λ ′ µ ′ (mod p t +1 ) .Proof: As in the proof of Corollary 3.4 we may assume without loss of generality that n ≥ w ′ . Then | λ ′ | = | λ | ≤ n . It follows from Proposition 3.3 that m λ ′ = ψ λ ′ ( e , . . . , e n )for some ψ λ ′ ∈ R t . Using induction on k we see that for 1 ≤ i ≤ n and k ≥ e i ( X p j , . . . , X p j n ) p k ≡ e i ( X , . . . , X n ) p j + k (mod p k +1 ) . Since ψ λ ′ ∈ R t it follows that m λ ( X , . . . , X n ) = m λ ′ ( X p j , . . . , X p j n )= ψ λ ′ ( e ( X p j , . . . , X p j n ) , . . . , e n ( X p j , . . . , X p j n )) ≡ ψ λ ′ ( e ( X , . . . , X n ) p j , . . . , e n ( X , . . . , X n ) p j ) (mod p t +1 ) . We also have m λ = ψ λ ( e , . . . , e n ). Hence by the fundamental theorem of symmetricpolynomials we get ψ λ ( X , . . . , X n ) ≡ ψ λ ′ ( X p j , . . . , X p j n ) (mod p t +1 ) . w ′ ≤ n the parts of µ ′ and µ are all ≤ n . Therefore the formula for ψ λ ′ givenby Theorem 2.1 includes the term d µ ′ λ ′ X µ ′ X µ ′ . . . X µ ′ h , and the formula for ψ λ includesthe term d µλ X µ X µ . . . X µ pjh = d µλ X p j µ ′ X p j µ ′ . . . X p j µ ′ h . It follows that d µλ ≡ d µ ′ λ ′ (mod p t +1 ), and hence that d λµ ≡ d λ ′ µ ′ (mod p t +1 ). (cid:3) Let
L/K be a totally ramified extension of degree n = up ν , with p ∤ u . Let σ , . . . , σ n be the K -embeddings of L into K sep . Let 1 ≤ h ≤ n and recall that E h : L → K isdefined by E h ( α ) = e h ( σ ( α ) , . . . , σ n ( α )) for α ∈ L . In this section we define a function γ h : Z → Z such that for r ∈ Z we have E h ( M rL ) ⊂ M γ h ( r ) K . The function γ h willbe defined in terms of the indices of inseparability of the extension L/K . In the nextsection we show that O K · E h ( M rL ) = M γ h ( r ) K holds in certain cases.Let π L be a uniformizer for L and let f ( X ) = X n − c X n − + · · · + ( − n − c n − X + ( − n c n be the minimum polynomial of π L over K . Then c h = E h ( π L ). For k ∈ Z define v p ( k ) = min { v p ( k ) , ν } . For 0 ≤ j ≤ ν set i π L j = min { nv K ( c h ) − h : 1 ≤ h ≤ n, v p ( h ) ≤ j } = min { v L ( c h π n − hL ) : 1 ≤ h ≤ n, v p ( h ) ≤ j } − n. Then i π L j is either a nonnegative integer or ∞ . If char( K ) = p then i π L j must be finite,since L/K is separable. If i π L j is finite write i π L j = a j n − b j with 1 ≤ b j ≤ n . Then v K ( c b j ) = a j , v K ( c h ) ≥ a j for all h with 1 ≤ h < b j and v p ( h ) ≤ j , and v K ( c h ) ≥ a j + 1for all h with b j < h ≤ n and v p ( h ) ≤ j . Let e L = v L ( p ) denote the absolute ramificationindex of L . We define the j th index of inseparability of L/K to be i j = min { i π L j ′ + ( j ′ − j ) e L : j ≤ j ′ ≤ ν } . By Proposition 3.12 and Theorem 7.1 of [4], i j does not depend on the choice of π L .Furthermore, our definition of i j agrees with Definition 7.3 in [4] (see also [5, Remark 2.5];for the characteristic- p case see [2, pp. 232–233] and [3, § i ν < i ν − ≤ · · · ≤ i ≤ i < ∞ .2. If char( K ) = p then e L = ∞ , and hence i j = i π L j .3. Let m = v p ( i j ). If m ≤ j then i j = i m = i π L j = i π L m . If m > j then char( K ) = 0and i j = i π L m + ( m − j ) e L . 8 emma 4.1 Let ≤ h ≤ n and set j = v p ( h ) . Then v L ( c h ) ≥ i π L j + h , with equality ifand only if either i π L j = ∞ or i π L j < ∞ and h = b j .Proof: If i π L j = ∞ then we certainly have v L ( c h ) = ∞ . Suppose i π L j < ∞ . If b j < h ≤ n then v L ( c h ) = nv K ( c h ) ≥ n ( a j + 1), and hence v L ( c h ) ≥ na j + n > na j − b j + h = i π L j + h. If 1 ≤ h < b j then v L ( c h ) ≥ na j > na j − b j + h = i π L j + h. Finally, we observe that v L ( c b j ) = na j = i π L j + b j . (cid:3) For a partition λ = { λ , . . . , λ k } whose parts satisfy λ i ≤ n for 1 ≤ i ≤ k define c λ = c λ c λ . . . c λ k . Proposition 4.2
Let w ≥ and let λ = { λ , . . . , λ k } be a partition of w whose partssatisfy λ i ≤ n . Choose q to minimize v p ( λ q ) and set t = v p ( λ q ) . Then v L ( c λ ) ≥ i π L t + w .If v L ( c λ ) = i π L t + w and i π L t < ∞ then λ q = b t and λ i = b ν = n for all i = q .Proof: If i π L t = ∞ then v L ( c λ q ) = ∞ , and hence v L ( c λ ) = ∞ . Suppose i π L t < ∞ . ByLemma 4.1 we have v L ( c λ q ) ≥ i π L t + λ q , and v L ( c λ i ) ≥ λ i for i = q . Hence v L ( c λ ) ≥ i π L t + w ,with equality if and only if v L ( c λ q ) = i π L t + λ q and v L ( c λ i ) = λ i for i = q . It follows fromLemma 4.1 that these conditions hold if and only if λ q = b t and λ i = b ν for all i = q . (cid:3) Proposition 4.3
Let w ≥ , let µ be a partition of w with h ≤ n parts, and set j = v p ( h ) . Let λ = { λ , . . . , λ k } be a partition of w whose parts satisfy λ i ≤ n , choose q to minimize v p ( λ q ) , and set t = v p ( λ q ) . Then(a) v L ( d λµ c λ ) ≥ i j + w .(b) Suppose v L ( d λµ c λ ) = i j + w . Then i π L t is finite, λ q = b t , and λ i = n for all i = q .Proof: (a) Suppose t ≥ j . Then by Corollary 3.4 we have v p ( d λµ ) ≥ t − j . Hence byProposition 4.2 we get v L ( d λµ c λ ) ≥ ( t − j ) e L + i π L t + w ≥ i j + w. Suppose t < j . Using Proposition 4.2 we get v L ( d λµ c λ ) ≥ v L ( c λ ) ≥ i π L t + w ≥ i t + w ≥ i j + w. (b) If v L ( d λµ c λ ) = i j + w then all the inequalities above are equalities. In either caseit follows that i π L t is finite and v L ( c λ ) = i π L t + w . Therefore by Proposition 4.2 we get λ q = b t and λ i = n for all i = q . (cid:3) We now apply some of the results of section 2 to our field extension
L/K . For α ∈ L let M µ ( α ) = m µ ( σ ( α ) , . . . , σ n ( α )). 9 roposition 4.4 Let r ≥ and let α ∈ M rL . Choose a power series φ ( X ) = a r X r + a r +1 X r +1 + . . . with coefficients in O K such that α = φ ( π L ) . Then E h ( α ) = X µ a µ a µ . . . a µ h M µ ( π L ) , where the sum ranges over all partitions µ = { µ , . . . , µ h } with h parts such that µ i ≥ r for ≤ i ≤ h .Proof: This follows from (2.1) by setting X i = σ i ( π L ) and letting a j ∈ O K . (cid:3) Proposition 4.5
Let n ≥ , let w ≥ , and let µ be a partition of w with at most n parts. Then M µ ( π L ) = X λ d λµ c λ , where the sum is over all partitions λ = { λ , . . . , λ k } of w such that λ i ≤ n for ≤ i ≤ k .Proof: This follows from Theorem 2.1 by setting X i = E i ( π L ) = c i . (cid:3) Let 1 ≤ h ≤ n and recall that we defined g h : Z → Z by setting g h ( r ) = s , where s is the largest integer such that E h ( M rL ) ⊂ M sK . Theorem 4.6
Let
L/K be a totally ramified extension of degree n = up ν , with p ∤ u .Let r ∈ Z , let ≤ h ≤ n , and set j = v p ( h ) . Then E h ( M rL ) ⊂ M ⌈ ( i j + hr ) /n ⌉ K g h ( r ) ≥ (cid:24) i j + hrn (cid:25) . Proof:
Let π K be a uniformizer for K . Then for t ∈ Z we have E h ( M nt + rL ) = E h ( π tK · M rL ) = π htK · E h ( M rL ) (4.1) (cid:24) i j + h ( nt + r ) n (cid:25) = ht + (cid:24) i j + hrn (cid:25) . (4.2)Therefore it suffices to prove the theorem in the cases with 1 ≤ r ≤ n . By Proposition 4.4each element of E h ( M rL ) is an O K -linear combination of terms of the form M µ ( π L ),where µ is a partition with h parts, all ≥ r . Fix one such partition µ and set w =Σ( µ ); then w ≥ hr . Using Proposition 4.5 we can express M µ ( π L ) as a sum of terms d λµ c λ , where λ = { λ , λ , . . . , λ k } is a partition of w into parts which are ≤ n . ByProposition 4.3(a) we get v L ( d λµ c λ ) ≥ i j + w ≥ i j + hr . Since d λµ c λ ∈ K it follows that v K ( d λµ c λ ) ≥ ⌈ ( i j + hr ) /n ⌉ . Therefore we have v K ( M µ ( π L )) ≥ ⌈ ( i j + hr ) /n ⌉ , and hence E h ( M rL ) ⊂ M ⌈ ( i j + hr ) /n ⌉ K . (cid:3) Equality
In this section we show that in some special cases we have O K · E h ( M rL ) = M ⌈ ( i j + hr ) /n ⌉ K ,where j = v p ( h ). This is equivalent to showing that g h ( r ) = ⌈ ( i j + hr ) /n ⌉ holds inthese cases. In particular, we prove that if the residue field K of K is large enough then g p j ( r ) = ⌈ ( i j + rp j ) /n ⌉ for 0 ≤ j ≤ ν . To prove that g h ( r ) = ⌈ ( i j + hr ) /n ⌉ holds for all r ∈ Z , by Theorem 4.6 it suffices to show the following: Let r satisfy (cid:24) i j + hrn (cid:25) < (cid:24) i j + h ( r + 1) n (cid:25) . (5.1)Then there is α ∈ M rL such that v K ( E h ( α )) = ⌈ ( i j + hr ) /n ⌉ . By (4.1) and (4.2) it’senough to prove this for r such that 1 ≤ r ≤ n .Once again we let π L be a uniformizer for L whose minimum polynomial over K is f ( X ) = X n − c X n − + · · · + ( − n − c n − X + ( − n c n . Theorem 5.1
Let
L/K be a totally ramified extension of degree n = up ν , with p ∤ u .Let j be an integer such that ≤ j ≤ ν and v p ( i j ) ≥ j . Then O K · E p j ( M rL ) = M ⌈ ( i j + rp j ) /n ⌉ K g p j ( r ) = (cid:24) i j + rp j n (cid:25) . Proof:
Set m = v p ( i j ). Then i j = ( m − j ) e L + i π L m . In particular, if char( K ) = p then m = j and i j = i m = i π L m . We can write i π L m = an − b with 1 ≤ b ≤ n and v p ( b ) = m .Since j ≤ m there is b ′ ∈ Z such that b = b ′ p j . Let r ∈ Z and set r = b ′ + r up ν − j .Then i j + rp j = ( m − j ) e L + an + r n. (5.2)Therefore we have (cid:24) i j + rp j n (cid:25) = ( m − j ) e K + a + r (cid:24) i j + ( r + 1) p j n (cid:25) = ( m − j ) e K + a + r + 1 , with e K = v K ( p ) = e L /n . It follows that the only values of r in the range 1 ≤ r ≤ n satisfying (5.1) are of the form r = b ′ + r up ν − j with 0 ≤ r < p j . Therefore it sufficesto prove that v K ( E p j ( π rL )) = ( m − j ) e K + a + r holds for these values of r .Let µ be the partition of rp j consisting of p j copies of r . Then E p j ( π rL ) = M µ ( π L ),so it follows from Proposition 4.5 that E p j ( π rL ) = X λ d λµ c λ , (5.3)11here the sum is over all partitions λ = { λ , . . . , λ k } of rp j such that λ i ≤ n for1 ≤ i ≤ k . It follows from Proposition 4.3(a) that v L ( d λµ c λ ) ≥ i j + rp j . Suppose v L ( d λµ c λ ) = i j + rp j . Then by Proposition 4.3(b) we see that λ has at most one elementwhich is not equal to n . Since Σ( λ ) = rp j = b + r n , and the elements of λ are ≤ n , itfollows that λ = κ , where κ is the partition of rp j which consists of 1 copy of b and r copies of n . Since E p j ( π rL ) ∈ K and d κµ c κ ∈ K it follows from (5.3) and (5.2) that E p j ( π rL ) ≡ d κµ c κ (mod M ( m − j ) e K + a + r +1 K ) . (5.4)Let κ ′ be the partition of r consisting of 1 copy of b ′ and r copies of up ν − j , and let µ ′ be the partition of r consisting of 1 copy of r . Then κ = p j · κ ′ and µ = p j ∗ µ ′ . Since v p ( b ′ ) = m − j it follows from Proposition 3.5 that d κµ ≡ d κ ′ µ ′ (mod p m − j +1 ). Suppose m < ν . Then b < n , so b ′ = up ν − j . Hence by Proposition 2.4 we get d κ ′ µ ′ = ( − r + r +1 r .Since r = b ′ + r up ν − j and v p ( b ′ ) = m − j this implies v p ( d κ ′ µ ′ ) = v p ( r ) = m − j .Suppose m = ν . Then b = n and b ′ = p − j b = up ν − j , so κ ′ consists of r + 1 copies of up ν − j . Since gcd( up ν − j , r ) = up ν − j and gcd( r + 1 ,
1) = 1, by Proposition 2.6 we get d κ ′ µ ′ = ( − r + r +1 up ν − j . Hence v p ( d κ ′ µ ′ ) = ν − j = m − j holds in this case as well. Since d κµ ≡ d κ ′ µ ′ (mod p m − j +1 ) it follows that v p ( d κ ′ µ ′ ) = m − j . Therefore v K ( d κµ c κ ) =( m − j ) e K + a + r . Using (5.4) we conclude that v K ( E p j ( π rL )) = ( m − j ) e K + a + r . (cid:3) Theorem 5.2
Let
L/K be a totally ramified extension of degree n = up ν , with p ∤ u .Let j be an integer such that ≤ j ≤ ν and v p ( i j ) < j . Set m = v p ( i j ) and assume that | K | > p m . Then O K · E p j ( M rL ) = M ⌈ ( i j + rp j ) /n ⌉ K g p j ( r ) = (cid:24) i j + rp j n (cid:25) . Proof:
Since m < j we have i m = i j = i π L j . Therefore i j = an − b for some a, b such that1 ≤ b < n and v p ( b ) = m . Hence b = b ′ p j + b ′′ p m for some b ′ , b ′′ such that 0 < b ′′ < p j − m and p ∤ b ′′ . Let r ∈ Z and set r = b ′ + r up ν − j . Then i j + rp j = an + r n − b ′′ p m , (5.5)so we have (cid:24) i j + rp j n (cid:25) = a + r + (cid:24) − b ′′ p m n (cid:25) = a + r (cid:24) i j + ( r + 1) p j n (cid:25) = a + r + (cid:24) p j − b ′′ p m n (cid:25) = a + r + 1 . It follows that the only values of r in the range 1 ≤ r ≤ n satisfying (5.1) are of theform r = b ′ + r up ν − j with 0 ≤ r < p j . It suffices to prove that for every such r thereis β ∈ O K such that v K ( E p j ( π rL + βπ r + b ′′ L )) = a + r .12et η ( X ) = E p j ( π rL + Xπ r + b ′′ L ). We need to show that there is β ∈ O K such that v K ( η ( β )) = a + r . It follows from Proposition 4.4 that η ( X ) is a polynomial in X ofdegree at most p j , with coefficients in O K . For 0 ≤ ℓ ≤ p j let µ ℓ be the partition of rp j + ℓb ′′ consisting of p j − ℓ copies of r and ℓ copies of r + b ′′ . By Proposition 4.4 thecoefficient of X ℓ in η ( X ) is equal to M µ ℓ ( π L ). By Proposition 4.5 we have M µ ℓ ( π L ) = X λ d λµ ℓ c λ , (5.6)where the sum is over all partitions λ = { λ , . . . , λ k } of rp j + ℓb ′′ such that λ i ≤ n for1 ≤ i ≤ k . Using Proposition 4.3(a) and (5.5) we get v L ( d λµ ℓ c λ ) ≥ i j + rp j + ℓb ′′ = ( a + r ) n + ( ℓ − p m ) b ′′ (5.7) > ( a + r − n. Since d λµ ℓ c λ ∈ K it follows that d λµ ℓ c λ ∈ M a + r K . Therefore by Proposition 4.5 we have M µ ℓ ( π L ) ∈ M a + r K .Suppose v K ( d λµ ℓ c λ ) = a + r . Then v L ( d λµ ℓ c λ ) = ( a + r ) n , so by (5.7) we get ℓ ≤ p m .Hence for p m < ℓ ≤ p j we have M µ ℓ ( π L ) ∈ M a + r +1 K . Let w = b + r n = rp j + b ′′ p m andlet µ = µ p m be the partition of w consisting of p m copies of r + b ′′ and p j − p m copies of r . Then the coefficient of X p m in η ( X ) is M µ ( π L ). Let κ be the partition of w consistingof 1 copy of b and r copies of n . Suppose λ is a partition of w with parts ≤ n suchthat v K ( d λµ c λ ) = a + r . Since ( a + r ) n = i j + w it follows from Proposition 4.3(b)that λ has at most one element which is not equal to n . Since Σ( λ ) = b + r n , and theelements of λ are ≤ n , it follows that λ = κ . Hence by (5.6) we have M µ ( π L ) ≡ d κµ c κ (mod M a + r +1 K ) . (5.8)Set w ′ = b ′ p j − m + b ′′ + r up ν − m = rp j − m + b ′′ . Let κ ′ be the partition of w ′ consistingof 1 copy of b ′ p j − m + b ′′ and r copies of up ν − m , and let µ ′ be the partition of w ′ consistingof 1 copy of r + b ′′ and p j − m − r . Then κ = p m · κ ′ and µ = p m ∗ µ ′ , soby Proposition 3.5 we have d κµ ≡ d κ ′ µ ′ (mod p ). Let Γ be a cycle digraph which hasan admissible ( κ ′ , µ ′ )-tiling. Suppose Γ has more than one component. Since Γ hasa κ -tiling, Γ has at least one component Γ such that up ν − m divides | V (Γ ) | . Thus | V (Γ ) | = k · up ν − m for some k such that 1 ≤ k ≤ r . Let κ ′ be the submultiset of κ ′ consisting of k copies of up ν − m . Then κ ′ is the unique submultiset of κ ′ such that Γ has a κ ′ -tiling. Furthermore there is a unique submultiset µ ′ of µ ′ such that Γ has a µ ′ -tiling.Suppose r does not divide kup ν − m . Then there is ℓ ≥ µ ′ consists of 1copy of r + b ′′ together with ℓ copies of r . By Proposition 2.3 we have η κ ′ µ ′ (Γ ) = up ν − m .Let Γ be the complement of Γ in Γ, let κ ′ = κ ′ r κ ′ , and let µ ′ = µ ′ r µ ′ . Since Γ has no cycle of length | V (Γ ) | = b ′′ + ( ℓ + 1) r we have η κ ′ µ ′ (Γ) = η κ ′ µ ′ (Γ ) η κ ′ µ ′ (Γ ).Hence η κ ′ µ ′ (Γ) is divisible by p in this case.13n the other hand, suppose r divides kup ν − m . Then there is ℓ ≥ µ ′ consists of ℓ copies of r . It follows that k · up ν − m = ℓ · r . Let ( S, T ) be an admissible( κ ′ , µ ′ )-tiling of Γ and let ( S , T ) be the restriction of ( S, T ) to Γ . Then ( S , T ) isa ( κ ′ , µ ′ )-tiling of Γ . By Proposition 2.5(b) the automorphism group of (Γ , S , T )has order gcd( k, ℓ ). Since Aut(Γ , S , T ) is isomorphic to a subgroup of Aut(Γ , S, T ),it follows that gcd( k, ℓ ) divides | Aut(Γ , S, T ) | . Therefore the assumption that ( S, T ) isadmissible implies that gcd( k, ℓ ) = 1. Hence k | r and ℓ | up ν − m , so there is q ∈ Z with r = kq and up ν − m = ℓq . By Proposition 2.5(a) the number of isomorphism classes of( κ ′ , µ ′ )-tilings of Γ is η κ ′ µ ′ (Γ ) = gcd( up ν − m , r ) = gcd( ℓq, kq ) = q. If p | q then as above we deduce that η κ ′ µ ′ (Γ) is divisible by p . On the other hand, if p ∤ q then q | u ; in particular, q ≤ u . Since k ≤ r we get r up ν − j + b ′ = r = kq ≤ r u ,a contradiction. By combining the two cases we find that if Γ has more than onecomponent then η κ ′ µ ′ (Γ) is divisible by p .Finally, suppose that Γ consists of a single cycle of length w ′ . Then by Proposition 2.2we have η κ ′ , µ ′ (Γ) = w ′ . Hence by (2.2) we get d κµ ≡ d κ ′ µ ′ ≡ ± η κ ′ µ ′ (Γ) ≡ ± w ′ (mod p ) . Since w ′ ≡ b ′′ (mod p ) it follows that p ∤ d κµ . Hence by (5.8) we get v K ( M µ ( π L )) = v K ( c κ ) = a + r . Let π K be a uniformizer for K and set φ ( X ) = π − a − r K η ( X ). Then φ ( X ) ∈ O K [ X ]. Let φ ( X ) be the image of φ ( X ) in K [ X ]. We have shown that φ ( X ) has degree p m . Since | K | > p m there is β ∈ K such that φ ( β ) = 0. Let β ∈ O K be a lifting of β . Then φ ( β ) ∈ O × K . It follows that v K ( E p j ( π rL + βπ r + b ′′ L )) = v K ( η ( β )) = a + r . We conclude that O K · E p j ( M rL ) = M ⌈ ( i j + rp j ) /n ⌉ K . (cid:3) Remark 5.3
Theorems 5.1 and 5.2 together imply that if K is sufficiently large then g p j ( r ) = ⌈ ( i j + rp j ) /n ⌉ for 0 ≤ j ≤ ν . This holds for instance if | K | ≥ p ν . Remark 5.4
Let
L/K be a totally ramified separable extension of degree n = up ν .The different M d L of L/K is defined by letting d be the largest integer such that E ( M − d L ) ⊂ O K . For 1 ≤ j ≤ ν one can define higher order analogs M d j L of thedifferent by letting d j be the largest integer such that E p j ( M − d j L ) ⊂ O K . An argumentsimilar to the proof of Proposition 1.2 shows that O K · E p j ( M rL ) = M ⌊ p j ( d j + r ) /n ⌋ K . This generalizes Proposition 1.2, which is equivalent to the case j = 0 of this formula.By Proposition 3.18 of [4], the valuation of the different of L/K is d = i + n −
1. Using14heorems 5.1 and 5.2 we find that, if K is sufficiently large, d j is the largest integersuch that ⌈ ( i j − d j p j ) /n ⌉ ≥
0. Hence d j = ⌊ ( i j + n − /p j ⌋ for 0 ≤ j ≤ ν . Example 5.5
Let K = F (( t )) and let L be an extension of K generated by a root π L ofthe Eisenstein polynomial f ( X ) = X + tX + tX + t . Then the indices of inseparabilityof L/K are i = 3, i = i = 2, and i = 0. Since ⌈ ( i + 2 · / ⌉ = 1, the formula inTheorem 5.2 would imply O K · E ( M L ) = M K . We claim that E ( M L ) ⊂ M K .Let α ∈ M L and write α = a π L + a π L + . . . , with a i ∈ F . It follows fromPropositions 4.4 and 4.5 that E ( α ) is a sum of terms of the form a µ d λµ c λ , where λ is apartition whose parts are ≤ µ is a partition with 4 parts such that Σ( λ ) = Σ( µ ).We are interested only in those terms with K -valuation 1. We have v K ( c λ ) ≥ λ is one of { } , { } , or { } . If λ = { } then 2 | d λµ for any µ by Corollary 3.4. If λ = { } and µ = { , , , } then d λµ = 6 by Proposition 2.4. If λ = { } and µ = { , , , } then a computation based on (2.2) shows that d λµ = 9. If λ = { } and µ = { , , , } then d λµ = − E ( α ) ≡ a a t + a a t (mod M K ) . Since a , a ∈ F we have a a + a a = 0. Therefore E ( α ) ∈ M K . Since this holdsfor every α ∈ M L we get E ( M L ) ⊂ M K . This shows that Theorem 5.2 does not holdwithout the assumption about the size of K .The following result shows that g h ( r ) = ⌈ ( i j + hr ) /n ⌉ does not hold in general, evenif we assume that the residue field of K is large. It also suggests that there may not bea simple criterion for determining when g h ( r ) = ⌈ ( i j + hr ) /n ⌉ does hold. Proposition 5.6
Let
L/K be a totally ramified extension of degree n , with p ∤ n . Let r ∈ Z and ≤ h ≤ n be such that n | hr . Set s = hr/n , u = gcd( r, n ) , and v = gcd( h, s ) .Then g h ( r ) = ⌈ ( i + hr ) /n ⌉ = s if and only if p does not divide the binomial coefficient (cid:18) uv (cid:19) . In particular, if u < v then g h ( r ) > s .Proof: Since
L/K is tamely ramified we have ν = 0, i = 0, and (cid:24) i + hrn (cid:25) = (cid:24) hrn (cid:25) = s. It follows from Theorem 4.6 that g h ( r ) ≥ s . If r ′ = nt + r then s ′ = hr ′ /n = ht + s , u ′ = gcd( r ′ , n ) = u , and v ′ = gcd( h, s ′ ) = v . Hence by (4.1) it suffices to prove theproposition in the cases with 1 ≤ r ≤ n .Suppose p does not divide (cid:18) uv (cid:19) . To prove g h ( r ) = s it suffices to show that v K ( E h ( π rL )) = s . Let µ be the partition of hr consisting of h copies of r . Then E h ( π rL ) = M µ ( π L ), so it follows from Proposition 4.5 that E h ( π rL ) = X λ d λµ c λ , (5.9)15here the sum is over all partitions λ = { λ , . . . , λ k } of hr such that λ i ≤ n for 1 ≤ i ≤ k .Let κ be the partition of hr = sn consisting of s copies of n and let λ be a partitionof hr whose parts are ≤ n . Then by Proposition 4.3(a) we have v L ( d κµ c λ ) ≥ hr = sn .Furthermore, if v L ( d κµ c λ ) = hr then by Proposition 4.3(b) we have λ = κ . Hence by(5.9) we get E h ( π rL ) ≡ d κµ c κ (mod M s +1 K ) . By Proposition 2.6 we have d κµ = ± (cid:18) uv (cid:19) . Since p ∤ (cid:18) uv (cid:19) and v K ( c κ ) = s it follows that v K ( E h ( π rL )) = s . Therefore g h ( r ) = s .Suppose p divides (cid:18) uv (cid:19) . By Proposition 4.4, each element of E h ( M rL ) is an O K -linearcombination of terms of the form M ν ( π L ) where ν is a partition with h parts, all ≥ r .Fix one such partition ν and set w = Σ( ν ); then w ≥ hr = sn . By Proposition 4.5 wecan express M ν ( π L ) as a sum of terms of the form d λν c λ , where λ = { λ , λ , . . . , λ k } is apartition of w into parts which are ≤ n . By Proposition 4.3(a) we have v L ( d λν c λ ) ≥ w ≥ sn . Suppose v L ( d λν c λ ) = sn . Then w = sn , and by Proposition 4.3(b) we see that λ consists of k copies of n . It follows that kn = w = sn , and hence that k = s . Therefore λ = κ . Since Σ( ν ) = w = kn = hr we get ν = µ . Since d κµ = ± (cid:18) uv (cid:19) and p divides (cid:18) uv (cid:19) we have v L ( d κµ c κ ) > v L ( c κ ) = sn , a contradiction. Hence v L ( d λν c λ ) > sn holds inall cases. Since d λν c λ ∈ K we get v K ( d λν c λ ) ≥ s + 1. It follows that E h ( M rL ) ⊂ M s +1 K ,and hence that g h ( r ) ≥ s + 1. (cid:3) References [1] I. B. Fesenko and S. V. Vostokov,