Perturbing Eisenstein polynomials over local fields

TL;DR

This paper studies how small perturbations of uniformizers in totally ramified extensions over local fields affect their minimal polynomials, providing improved congruences that extend previous results.

Contribution

It derives new congruences for the coefficients of perturbed minimal polynomials, extending Krasner's work on Eisenstein polynomials over local fields.

Findings

Provides explicit congruences for polynomial coefficients after perturbation.

Extends Krasner's results with more general and refined conditions.

Enhances understanding of ramification and polynomial stability in local field extensions.

Abstract

Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension. Let $\pi_L$ be a uniformizer for $L$ and let $f(X)$ be the minimum polynomial for $\pi_L$ over $K$. Suppose $\tilde{\pi}_L$ is another uniformizer for $L$ such that $\tilde{\pi}_L\equiv\pi_L+r\pi_L^{\ell+1} \pmod{\pi_L^{\ell+2}}$ for some $\ell\ge1$ and $r\in O_K$. Let $\tilde{f}(X)$ be the minimum polynomial for $\tilde{\pi}_L$ over $K$. In this paper we give congruences for the coefficients of $\tilde{f}(X)$ in terms of $r$ and the coefficients of $f(X)$. These congruences improve and extend work of Krasner.

Equations

$$f(X)=X^n-c_1{X}^{n-1}+\cdots +(-1{)}^{n-1}{c}_{n-1}X+(-1{)}^n{c}_n$$

$$\tilde{f}(X)=X^n-{\tilde{c}}_1{X}^{n-1}+\cdots +(-1{)}^{n-1}{\tilde{c}}_{n-1}X+(-1{)}^n{\tilde{c}}_n$$

$$d_{\mathbf{\lambda }\mathbf{\mu }}=(-1{)}^{|\mathbf{\lambda }|+|\mathbf{\mu }|}\cdot \sum _{\Gamma }\mathrm{sgn}(\Gamma ){\eta }_{\mathbf{\lambda }\mathbf{\mu }}(\Gamma ),$$

$${\psi }_{\mathbf{\mu }}(X_1,\dots ,X_n)=\sum _{\mathbf{\lambda }}{d}_{\mathbf{\lambda }\mathbf{\mu }}\cdot X_{{\lambda }_1}{X}_{{\lambda }_2}\dots X_{{\lambda }_k},$$

$$d_{\mathbf{\lambda }\mathbf{\mu }}=\{\begin{array}{ll}(-1{)}^{r+s+w+1}w& \text{if }b\nmid c\text{ or }sb<c,\\ (-1{)}^{r+s+w+1}(w-ab)& \text{if b∣c and }sb\ge c.\end{array}$$

$${\eta }_{\mathbf{\lambda }\mathbf{\mu }}(\Gamma )={\eta }_{{\mathbf{\lambda }}_1{\mathbf{\mu }}_1}({\Gamma }_1)\cdot {\eta }_{{\mathbf{\lambda }}_2{\mathbf{\mu }}_2}({\Gamma }_2)=ba.$$

$$d_{\mathbf{\lambda }\mathbf{\mu }}=(-1{)}^{(r+1)+(s+1)}\cdot (-1{)}^{w-1}w.$$

$$d_{\mathbf{\lambda }\mathbf{\mu }}=(-1{)}^{(r+1)+(s+1)}((-1{)}^{w-1}w+(-1{)}^{w-2}ab).$$

$$f(X)=X^n-c_1{X}^{n-1}+\cdots +(-1{)}^{n-1}{c}_{n-1}X+(-1{)}^n{c}_n.$$

$$i_j^{{\pi }_L}$$

$$=\min \{v_L(c_h{\pi }_L^{n-h}):1\le h\le n,{\overline{v}}_p(h)\le j\}-n.$$

$$i_j=\min \{i_{j^{\prime }}^{{\pi }_L}+(j^{\prime }-j)e_L:j\le j^{\prime }\le \nu \}.$$

$${\varphi }_j(x)=\min \{{\tilde{\varphi }}_{j_0}(x):0\le j_0\le j\}.$$

$$f(X)$$

$$\tilde{f}(X)$$

$${\rho }_h(\ell )=\lceil \frac{{\varphi }_j(\ell )+h}{n}\rceil .$$

$$M_{\mathbf{\mu }}(\alpha )=m_{\mathbf{\mu }}({\sigma }_1(\alpha ),\dots ,{\sigma }_n(\alpha )).$$

$$E_h(\alpha )=e_h({\sigma }_1(\alpha ),\dots ,{\sigma }_n(\alpha )).$$

$$E_h({\tilde{\pi }}_L)=\sum _{\mathbf{\mu }}{r}_{{\mu }_1}{r}_{{\mu }_2}\dots r_{{\mu }_h}{M}_{\mathbf{\mu }}({\pi }_L),$$

$$M_{\mathbf{\mu }}({\pi }_L)=\sum _{\mathbf{\lambda }}{d}_{\mathbf{\lambda }\mathbf{\mu }}{c}_{\mathbf{\lambda }},$$

$${\tilde{c}}_h=E_h({\tilde{\pi }}_L)=\sum _{s=0}^h{M}_{{\mathbf{\mu }}_s}({\pi }_L)r^s=c_h+\sum _{s=1}^h{M}_{{\mathbf{\mu }}_s}({\pi }_L)r^s.$$

$$v_L(d_{\mathbf{\lambda }{\mathbf{\mu }}_s}{c}_{\mathbf{\lambda }})$$

$$\ge (t-m)v_L(p)+i_t^{{\pi }_L}+\ell s+h$$

$$\ge i_m+\ell p^m+h.$$

$$v_L(d_{\mathbf{\lambda }{\mathbf{\mu }}_s}{c}_{\mathbf{\lambda }})$$

$$\ge i_t^{{\pi }_L}+\ell s+h$$

$$\ge i_t+\ell p^m+h$$

$$\ge i_m+\ell p^m+h.$$

$${\kappa }_h(\ell )=\lceil \frac{{\varphi }_{\nu }(\ell )+h}{n}\rceil .$$

$${\tilde{c}}_h\equiv c_h+\sum _{m\in S_j}{g}_m{c}_n^{k-A_m}{c}_{b_m}{r}^{p^m}(\mathrm{mod}{\mathcal{M}}_K^{k+1}),$$

$$S_j$$

$$g_m$$

$$\hat{f}(X)=X^n-{\hat{c}}_1{X}^{n-1}+\cdots +(-1{)}^{n-1}{\hat{c}}_{n-1}X+(-1{)}^n{\hat{c}}_n$$

$$w_m={\varphi }_j(\ell )-i_m+h=(k-A_m)n+b_m,$$

Full text

1 Introduction

Let $K$ be a field which is complete with respect to a discrete valuation $v_{K}$. Let $\mathcal{O}_{K}$ be the ring of integers of $K$ and let $\mathcal{M}_{K}$ be the maximal ideal of $\mathcal{O}_{K}$. Assume that the residue field $\overline{K}=\mathcal{O}_{K}/\mathcal{M}_{K}$ of $K$ is a perfect field of characteristic $p$. Let $K^{sep}$ be a separable closure of $K$ and let $L/K$ be a finite totally ramified subextension of $K^{sep}/K$. Let $\pi_{L}$ be a uniformizer for $L$ and let

[TABLE]

be the minimum polynomial of $\pi_{L}$ over $K$. Let $\ell\geq 1$, let $r\in\mathcal{O}_{K}$, and let $\tilde{\pi}_{L}$ be another uniformizer for $L$ such that $\tilde{\pi}_{L}\equiv\pi_{L}+r\pi_{L}^{\ell+1}\pmod{\mathcal{M}_{L}^{\ell+2}}$. Let

[TABLE]

be the minimum polynomial of $\tilde{\pi}_{L}$ over $K$. In this paper we use the techniques developed in [6] to obtain congruences for the coefficients $\tilde{c}_{i}$ of $\tilde{f}(X)$ in terms of $r$ and the coefficients of $f(X)$.

Let $\phi_{L/K}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ be the Hasse-Herbrand function of $L/K$, as defined for instance in Chapter IV of [9]. For $1\leq h\leq n$ set $k_{h}=\lceil\phi_{L/K}(\ell)+\frac{h}{n}\rceil$. Krasner [7, p. 157] showed that for $1\leq h\leq n$ we have $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{k_{h}}}$. In Theorem 4.3 we prove that $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{k_{h}^{\prime}}}$ for certain integers $k_{h}^{\prime}$ such that $k_{h}^{\prime}\geq k_{h}$. Let $h$ be the unique integer such that $1\leq h\leq n$ and $n$ divides $n\phi_{L/K}(\ell)+h$. Krasner [7, p. 157] gave a formula for the congruence class modulo $\mathcal{M}_{K}^{k_{h}+1}$ of $\tilde{c}_{h}-c_{h}$. In Theorem 4.5 we give similar formulas for up to $\nu+1$ values of $h$, where $\nu=v_{p}(n)$.

Heiermann [3] gave formulas which are analogous to the results presented here. Let $S\subset\mathcal{O}_{K}$ be the set of Teichmul̈ler representatives for $\overline{K}$. Let $\pi_{K}$ be a uniformizer for $K$ and let $\mathcal{F}(X)$ be the unique power series with coefficients in $S$ such that $\pi_{K}=\pi_{L}^{n}\mathcal{F}(\pi_{L})$. Suppose $\tilde{\pi}_{L}$ is another uniformizer for $L$ such that $\tilde{\pi}_{L}\equiv\pi_{L}+r\pi_{L}^{\ell+1}\pmod{\mathcal{M}_{L}^{\ell+2}}$ for some $\ell\geq 1$ and $r\in S$. Let $\tilde{\mathcal{F}}$ be the series with coefficients in $S$ such that $\pi_{K}=\tilde{\pi}_{L}^{n}\tilde{\mathcal{F}}(\tilde{\pi}_{L})$. Using Theorem 4.6 of [3] one can compute certain coefficients of $\tilde{\mathcal{F}}$ in terms of $r$ and the coefficients of $\mathcal{F}$.

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 $\phi_{L/K}$. In Section 4 we prove our main results. In Section 5 we give some examples which illustrate how the theorems from Section 4 are applied.

2 Symmetric polynomials and cycle digraphs

Let $n\geq 1$, let $w\geq 1$, and let $\boldsymbol{\mu}$ be a partition of $w$. We view $\boldsymbol{\mu}$ as a multiset of positive integers such that the sum $\Sigma(\boldsymbol{\mu})$ of the elements of $\boldsymbol{\mu}$ is equal to $w$. The cardinality of $\boldsymbol{\mu}$ is denoted by $|\boldsymbol{\mu}|$. For $\boldsymbol{\mu}$ such that $|\boldsymbol{\mu}|\leq n$ we let $m_{\boldsymbol{\mu}}(X_{1},\dots,X_{n})$ be the monomial symmetric polynomial in $n$ variables associated to $\boldsymbol{\mu}$. For $1\leq h\leq n$ let $e_{h}(X_{1},\dots,X_{n})$ denote the elementary symmetric polynomial of degree $h$ in $n$ variables. By the fundamental theorem of symmetric polynomials there is a unique polynomial $\psi_{\boldsymbol{\mu}}\in\mathbb{Z}[X_{1},\dots,X_{n}]$ such that $m_{\boldsymbol{\mu}}=\psi_{\boldsymbol{\mu}}(e_{1},\dots,e_{n})$. In this section we use a theorem of Kulikauskas and Remmel [8] to compute certain coefficients of $\psi_{\boldsymbol{\mu}}$.

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 $\Gamma$ is a cycle digraph if it is a disjoint union of finitely many directed cycles of length $\geq 1$. We denote the vertex set of $\Gamma$ by $V(\Gamma)$, and we define the sign of $\Gamma$ to be $\mathrm{sgn}(\Gamma)=(-1)^{w-c}$, where $w=|V(\Gamma)|$ and $c$ is the number of cycles that make up $\Gamma$.

Let $\Gamma$ be a cycle digraph with $w\geq 1$ vertices and let $\boldsymbol{\lambda}$ be a partition of $w$. A $\boldsymbol{\lambda}$-tiling of $\Gamma$ is a set $S$ of subgraphs of $\Gamma$ such that

Each $\gamma\in S$ is a directed path of length $\geq 0$. 2. 2.

The collection $\{V(\gamma):\gamma\in S\}$ forms a partition of the set $V(\Gamma)$. 3. 3.

The multiset $\{|V(\gamma)|:\gamma\in S\}$ is equal to $\boldsymbol{\lambda}$.

Let $\boldsymbol{\mu}$ be another partition of $w$. A $(\boldsymbol{\lambda},\boldsymbol{\mu})$-tiling of $\Gamma$ is an ordered pair $(S,T)$, where $S$ is a $\boldsymbol{\lambda}$-tiling of $\Gamma$ and $T$ is a $\boldsymbol{\mu}$-tiling of $\Gamma$. Let $\Gamma^{\prime}$ be another cycle digraph with $w$ vertices and let $(S^{\prime},T^{\prime})$ be a $(\boldsymbol{\lambda},\boldsymbol{\mu})$-tiling of $\Gamma^{\prime}$. An isomorphism from $(\Gamma,S,T)$ to $(\Gamma^{\prime},S^{\prime},T^{\prime})$ is an isomorphism of digraphs $\theta:\Gamma\rightarrow\Gamma^{\prime}$ which carries $S$ onto $S^{\prime}$ and $T$ onto $T^{\prime}$. Say that the $(\boldsymbol{\lambda},\boldsymbol{\mu})$-tilings $(S,T)$ and $(S^{\prime},T^{\prime})$ of $\Gamma$ are isomorphic if there exists an isomorphism from $(\Gamma,S,T)$ to $(\Gamma,S^{\prime},T^{\prime})$. Say that $(S,T)$ is an admissible $(\boldsymbol{\lambda},\boldsymbol{\mu})$-tiling of $\Gamma$ if $(\Gamma,S,T)$ has no nontrivial automorphisms. Let $\eta_{\boldsymbol{\lambda}\boldsymbol{\mu}}(\Gamma)$ denote the number of isomorphism classes of admissible $(\boldsymbol{\lambda},\boldsymbol{\mu})$-tilings of $\Gamma$.

Let $w\geq 1$ and let $\boldsymbol{\lambda},\boldsymbol{\mu}$ be partitions of $w$. Set

[TABLE]

where the sum is over all isomorphism classes of cycle digraphs $\Gamma$ with $w$ vertices. Since $\eta_{\boldsymbol{\mu}\boldsymbol{\lambda}}=\eta_{\boldsymbol{\lambda}\boldsymbol{\mu}}$ we have $d_{\boldsymbol{\mu}\boldsymbol{\lambda}}=d_{\boldsymbol{\lambda}\boldsymbol{\mu}}$. Kulikauskas and Remmel [8, Th. 1(ii)] proved the following:

Theorem 2.1

Let $n\geq 1$, let $w\geq 1$, and let $\boldsymbol{\mu}$ be a partition of $w$ with at most $n$ parts. Let $\psi_{\boldsymbol{\mu}}$ be the unique element of $\mathbb{Z}[X_{1},\dots,X_{n}]$ such that $m_{\boldsymbol{\mu}}=\psi_{\boldsymbol{\mu}}(e_{1},\dots,e_{n})$. Then

[TABLE]

where the sum is over all partitions $\boldsymbol{\lambda}=\{\lambda_{1},\dots,\lambda_{k}\}$ of $w$ such that $1\leq\lambda_{i}\leq n$ for $1\leq i\leq k$.

We now recall some formulas from [6] for computing values of $\eta_{\boldsymbol{\lambda}\boldsymbol{\mu}}(\Gamma)$.

Proposition 2.2

*Let $a,b,c,d,w$ be positive integers such that $a\not=c$, $b\not=d$, and let $r,s$ be nonnegative integers. Let $\Gamma$ be a directed cycle of length $w$.

(a) Suppose $w=ra=sb+d$. Let $\boldsymbol{\lambda}$ be the partition of $w$ consisting of $r$ copies of $a$, and let $\boldsymbol{\mu}$ be the partition of $w$ consisting of $s$ copies of $b$ and one copy of $d$. Then $\eta_{\boldsymbol{\lambda}\boldsymbol{\mu}}(\Gamma)=a$.

(b) Suppose $w=ra+c=sb+d$. Let $\boldsymbol{\lambda}$ be the partition of $w$ consisting of $r$ copies of $a$ and one copy of $c$, and let $\boldsymbol{\mu}$ be the partition of $w$ consisting of $s$ copies of $b$ and one copy of $d$. Then $\eta_{\boldsymbol{\lambda}\boldsymbol{\mu}}(\Gamma)=w$.*

*Proof: *Statement (a) follows from Proposition 2.5 of [6] if $s=0$, and from Proposition 2.3 of [6] if $s\geq 1$. Statement (b) follows from Proposition 2.2 of [6]. $\square$

Using these formulas we can compute $d_{\boldsymbol{\lambda}\boldsymbol{\mu}}$ in some cases.

Proposition 2.3

Let $a,b,c,d,w$ be positive integers such that $a\not=c$ and $b\not=d$. Let $r,s$ be nonnegative integers such that $w=ra+c=sb+d$ and $a>sb$. Let $\boldsymbol{\lambda}$ be the partition of $w$ consisting of $r$ copies of $a$ and 1 copy of $c$, and let $\boldsymbol{\mu}$ be the partition of $w$ consisting of $s$ copies of $b$ and 1 copy of $d$. Then

[TABLE]

*Proof: *Let $\Gamma$ be a cycle digraph which has an admissible $(\boldsymbol{\lambda},\boldsymbol{\mu})$-tiling. Suppose $\Gamma$ consists of a single cycle of length $w$. Then by Proposition 2.2(b) we have $\eta_{\boldsymbol{\lambda}\boldsymbol{\mu}}(\Gamma)=w$. Suppose $\Gamma$ has more than one cycle. Since $\Gamma$ has a $\boldsymbol{\mu}$-tiling, $\Gamma$ has a cycle $\Gamma_{1}$ such that $|V(\Gamma_{1})|\leq sb$. Since $a>sb$ and $\Gamma$ has a $\boldsymbol{\lambda}$-tiling, it follows that $|V(\Gamma_{1})|=c=mb$ for some $m$ such that $1\leq m\leq s$. Hence if $\Gamma$ has more than one cycle we must have $b\mid c$ and $c\leq sb$. Let $\boldsymbol{\lambda}_{1}$ be the partition of $c$ consisting of one copy of $c$ and let $\boldsymbol{\mu}_{1}$ be the partition of $c$ consisting of $m$ copies of $b$. Then every $\boldsymbol{\lambda}$-tiling of $\Gamma$ restricts to a $\boldsymbol{\lambda}_{1}$-tiling of $\Gamma_{1}$, and every $\boldsymbol{\mu}$-tiling of $\Gamma$ restricts to a $\boldsymbol{\mu}_{1}$-tiling of $\Gamma_{1}$. It follows from Proposition 2.2(a) that $\eta_{\boldsymbol{\lambda}_{1}\boldsymbol{\mu}_{1}}(\Gamma_{1})=b$.

Let $\Gamma_{2}$ be another cycle of $\Gamma$. Since $\Gamma$ has a $\boldsymbol{\lambda}$-tiling, $|V(\Gamma_{2})|\geq a>sb$. Hence every $\boldsymbol{\mu}$-tiling of $\Gamma$ restricts to a tiling of $\Gamma_{2}$ which includes a path $\delta$ with $|V(\delta)|=d$. Since $\boldsymbol{\mu}$ has only one part equal to $d$, it follows that $\Gamma=\Gamma_{1}\cup\Gamma_{2}$. Therefore we have $|V(\Gamma_{2})|=ra=(s-m)b+d$. Let $\boldsymbol{\lambda}_{2}$ be the partition of $ra$ consisting of $r$ copies of $a$ and let $\boldsymbol{\mu}_{2}$ be the partition of $(s-m)b+d=ra$ consisting of $s-m$ copies of $b$ and 1 copy of $d$. Then every $\boldsymbol{\lambda}$-tiling of $\Gamma$ restricts to a $\boldsymbol{\lambda}_{2}$-tiling of $\Gamma_{2}$, and every $\boldsymbol{\mu}$-tiling of $\Gamma$ restricts to a $\boldsymbol{\mu}_{2}$-tiling of $\Gamma_{2}$. It follows from Proposition 2.2(a) that $\eta_{\boldsymbol{\lambda}_{2}\boldsymbol{\mu}_{2}}(\Gamma_{2})=a$. Hence

[TABLE]

Suppose $b\nmid c$ or $c>sb$. Then it follows from the above that the only cycle digraph which has a $(\boldsymbol{\lambda},\boldsymbol{\mu})$-tiling consists of a single cycle of length $w$. Hence by (2.1) we get

[TABLE]

Suppose $b\mid c$ and $sb\geq c$. Then $c=mb$ with $1\leq m\leq s$. Hence there are two cycle digraphs which have a $(\boldsymbol{\lambda},\boldsymbol{\mu})$-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

[TABLE]

Hence the formula for $d_{\boldsymbol{\lambda}\boldsymbol{\mu}}$ given in the theorem holds in both cases. $\square$

We recall some results from [6] regarding the $p$-adic properties of the coefficients $d_{\boldsymbol{\lambda}\boldsymbol{\mu}}$. Let $w\geq 1$ and let $\boldsymbol{\lambda}$ be a partition of $w$. For $k\geq 1$ let $k*\boldsymbol{\lambda}$ be the partition of $kw$ which is the multiset sum of $k$ copies of $\boldsymbol{\lambda}$, and let $k\cdot\boldsymbol{\lambda}$ be the partition of $kw$ obtained by multiplying the parts of $\boldsymbol{\lambda}$ by $k$.

Proposition 2.4

Let $t\geq j\geq 0$, let $w^{\prime}\geq 1$, and set $w=w^{\prime}p^{t}$. Let $\boldsymbol{\lambda}^{\prime}$ be a partition of $w^{\prime}$ and set $\boldsymbol{\lambda}=p^{t}\cdot\boldsymbol{\lambda}^{\prime}$. Let $\boldsymbol{\mu}$ be a partition of $w$ such that there does not exist a partition $\boldsymbol{\mu}^{\prime}$ with $\boldsymbol{\mu}=p^{j+1}*\boldsymbol{\mu}^{\prime}$. Then $p^{t-j}$ divides $d_{\boldsymbol{\lambda}\boldsymbol{\mu}}$.

*Proof: *This is proved in Corollary 3.4 of [6]. $\square$

Proposition 2.5

Let $w^{\prime}\geq 1$, $j\geq 1$, and $t\geq 0$. Let $\boldsymbol{\lambda}^{\prime}$, $\boldsymbol{\mu}^{\prime}$ be partitions of $w^{\prime}$ such that the parts of $\boldsymbol{\lambda}^{\prime}$ are all divisible by $p^{t}$. Set $w=w^{\prime}p^{j}$, so that $\boldsymbol{\lambda}=p^{j}\cdot\boldsymbol{\lambda}^{\prime}$ and $\boldsymbol{\mu}=p^{j}*\boldsymbol{\mu}^{\prime}$ are partitions of $w$. Then $d_{\boldsymbol{\lambda}\boldsymbol{\mu}}\equiv d_{\boldsymbol{\lambda}^{\prime}\boldsymbol{\mu}^{\prime}}\pmod{p^{t+1}}$.

*Proof: *This is proved in Proposition 3.5 of [6]. $\square$

3 Indices of inseparability

Let $L/K$ be a totally ramified extension of degree $n=up^{\nu}$, with $p\nmid u$. Let $\pi_{L}$ be a uniformizer for $L$ whose minimum polynomial over $K$ is

[TABLE]

For $k\in\mathbb{Z}$ define $\overline{v}_{p}(k)=\min\{v_{p}(k),\nu\}$. For $0\leq j\leq\nu$ set

[TABLE]

Then $i_{j}^{\pi_{L}}$ is either a nonnegative integer or $\infty$; if $\operatorname{char}(K)=p$ then $i_{j}^{\pi_{L}}$ must be finite, since $L/K$ is separable. Let $e_{L}=v_{L}(p)$ denote the absolute ramification index of $L$. We define the $j$th index of inseparability of $L/K$ to be

[TABLE]

By Proposition 3.12 and Theorem 7.1 of [3], $i_{j}$ does not depend on the choice of $\pi_{L}$. Furthermore, our definition of $i_{j}$ agrees with Definition 7.3 in [3]; for the characteristic-$p$ case see also [1, pp. 232–233] and [2, §2]. Write $i_{j}=A_{j}n-b_{j}$ with $1\leq b_{j}\leq n$.

Remark 3.1

If $i_{j}^{\pi_{L}}$ is finite we can write $i_{j}^{\pi_{L}}=a_{j}n-b_{j}$ with $a_{j}\geq 1$ (see Section 4 of [6]). Thus if $i_{j}=i_{j^{\prime}}^{\pi_{L}}+(j^{\prime}-j)e_{L}$ then $A_{j}=a_{j^{\prime}}+(j^{\prime}-j)e_{K}$.

The following facts are easy consequences of the definitions:

$0=i_{\nu}<i_{\nu-1}\leq\dots\leq i_{1}\leq i_{0}<\infty$. 2. 2.

If $\operatorname{char}(K)=p$ then $i_{j}=i_{j}^{\pi_{L}}$. 3. 3.

Let $m=\overline{v}_{p}(i_{j})$. If $m\leq j$ then $i_{j}=i_{m}=i_{j}^{\pi_{L}}=i_{m}^{\pi_{L}}$. If $m>j$ then $\operatorname{char}(K)=0$ and $i_{j}=i_{m}^{\pi_{L}}+(m-j)e_{L}$.

Following [3, (4.4)], for $0\leq j\leq\nu$ we define functions $\tilde{\phi}_{j}:[0,\infty)\rightarrow[0,\infty)$ by $\tilde{\phi}_{j}(x)=i_{j}+p^{j}x$. The generalized Hasse-Herbrand functions $\phi_{j}:[0,\infty)\rightarrow[0,\infty)$ are then defined by

[TABLE]

Hence we have $\phi_{j}(x)\leq\phi_{j^{\prime}}(x)$ for $0\leq j^{\prime}\leq j$. Let $\phi_{L/K}:[0,\infty)\rightarrow[0,\infty)$ be the usual Hasse-Herbrand function. Then by Corollary 6.11 of [3] we have $\phi_{\nu}(x)=n\phi_{L/K}(x)$.

For a partition $\boldsymbol{\lambda}=\{\lambda_{1},\dots,\lambda_{k}\}$ whose parts satisfy $1\leq\lambda_{i}\leq n$ define $c_{\boldsymbol{\lambda}}=c_{\lambda_{1}}c_{\lambda_{2}}\dots c_{\lambda_{k}}$. The following is proved in Proposition 4.2 of [6].

Proposition 3.2

Let $w\geq 1$ and let $\boldsymbol{\lambda}=\{\lambda_{1},\dots,\lambda_{k}\}$ be a partition of $w$ whose parts satisfy $1\leq\lambda_{i}\leq n$. Choose $q$ to minimize $\overline{v}_{p}(\lambda_{q})$ and set $t=\overline{v}_{p}(\lambda_{q})$. Then $v_{L}(c_{\boldsymbol{\lambda}})\geq i_{t}^{\pi_{L}}+w$. If $v_{L}(c_{\boldsymbol{\lambda}})=i_{t}^{\pi_{L}}+w$ and $i_{t}^{\pi_{L}}<\infty$ then $\lambda_{q}=b_{t}$ and $\lambda_{i}=b_{\nu}=n$ for all $i\not=q$.

4 Perturbing $\pi_{L}$

In this section we prove our main theorems. We begin by applying the results of Section 2 to the totally ramified extension $L/K$. Write $[L:K]=n=up^{\nu}$ with $p\nmid u$. Let $\pi_{L}$, $\tilde{\pi}_{L}$ be uniformizers for $L$, with minimum polynomials over $K$ given by

[TABLE]

Let $1\leq h\leq n$ and set $j=\overline{v}_{p}(h)$. Define a function $\rho_{h}:\mathbb{N}\rightarrow\mathbb{N}$ by

[TABLE]

Let $\ell\geq 1$. We say $\tilde{f}\sim_{\ell}f$ if $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{\rho_{h}(\ell)}}$ for $1\leq h\leq n$. Thus $\sim_{\ell}$ is an equivalence relation on the set of minimum polynomials over $K$ for uniformizers of $L$.

Let $\sigma_{1},\dots,\sigma_{n}$ be the $K$-embeddings of $L$ into $K^{sep}$. For each partition $\boldsymbol{\mu}$ with at most $n$ parts define $M_{\boldsymbol{\mu}}:L\rightarrow K$ by

[TABLE]

For $1\leq h\leq n$ define $E_{h}:L\rightarrow K$ by

[TABLE]

Then $c_{h}=E_{h}(\pi_{L})$ and $\tilde{c}_{h}=E_{h}(\tilde{\pi}_{L})$.

Proposition 4.1

Let $\phi(X)=r_{1}X+r_{2}X^{2}+\cdots$ be a power series with coefficients in $\mathcal{O}_{K}$ such that $\tilde{\pi}_{L}=\phi(\pi_{L})$. Then for $1\leq h\leq n$ we have

[TABLE]

where the sum ranges over all partitions $\boldsymbol{\mu}=\{\mu_{1},\dots,\mu_{h}\}$ with $h$ parts.

*Proof: *This is a special case of Proposition 4.4 in [6]. $\square$

Proposition 4.2

Let $n\geq 1$, let $w\geq 1$, and let $\boldsymbol{\mu}$ be a partition of $w$ with at most $n$ parts. Then

[TABLE]

where the sum is over all partitions $\boldsymbol{\lambda}=\{\lambda_{1},\dots,\lambda_{k}\}$ of $w$ such that $1\leq\lambda_{i}\leq n$ for $1\leq i\leq k$.

*Proof: *This follows from Theorem 2.1 by setting $X_{i}=E_{i}(\pi_{L})=c_{i}$. $\square$

Let $\ell\geq 1$. Our first main result gives congruences between the coefficients of $f(X)$ and the coefficients of $\tilde{f}(X)$ under the assumption $\tilde{\pi}_{L}\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{\ell+1}}$.

Theorem 4.3

Let $\pi_{L}$, $\tilde{\pi}_{L}$ be uniformizers for $L$ and let $f(X)$, $\tilde{f}(X)$ be the minimum polynomials for $\pi_{L}$, $\tilde{\pi}_{L}$ over $K$. Suppose there are $\ell\geq 1$ and $\sigma\in\mathrm{Aut}_{K}(L)$ such that $\sigma(\tilde{\pi}_{L})\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{\ell+1}}$. Then $\tilde{f}\sim_{\ell}f$.

*Proof: *We first show that the theorem holds in the case where $\tilde{\pi}_{L}=\pi_{L}+r\pi_{L}^{\ell+1}$, with $r\in\mathcal{O}_{K}$. Let $1\leq h\leq n$ and set $j=\overline{v}_{p}(h)$. For $0\leq s\leq h$ let $\boldsymbol{\mu}_{s}$ be the partition of $\ell s+h$ consisting of $h-s$ copies of 1 and $s$ copies of $\ell+1$. Then by Proposition 4.1 we have

[TABLE]

To prove that $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{\rho_{h}(\ell)}}$ it’s enough to show that $v_{K}(M_{\boldsymbol{\mu}_{s}}(\pi_{L}))\geq\rho_{h}(\ell)$ for $1\leq s\leq h$. Therefore by Proposition 4.2 it suffices to show $v_{L}(d_{\boldsymbol{\lambda}\boldsymbol{\mu}_{s}}c_{\boldsymbol{\lambda}})\geq\phi_{j}(\ell)+h$ for all $1\leq s\leq h$ and all partitions $\boldsymbol{\lambda}$ of $\ell s+h$ whose parts are at most $n$.

Let $1\leq s\leq h$, set $j=\overline{v}_{p}(h)$, and set $m=\min\{j,\overline{v}_{p}(s)\}$. Then $m\leq j$ and $s\geq p^{m}$. Let $\boldsymbol{\lambda}=\{\lambda_{1},\dots,\lambda_{k}\}$ be a partition of $\ell s+h$ such that $1\leq\lambda_{i}\leq n$ for $1\leq i\leq k$. Choose $q$ to minimize $\overline{v}_{p}(\lambda_{q})$ and set $t=\overline{v}_{p}(\lambda_{q})$. By Proposition 3.2 we have $v_{L}(c_{\boldsymbol{\lambda}})\geq i_{t}^{\pi_{L}}+\ell s+h$. Suppose $m<t$. Then $m<\nu$, so we have $p^{m+1}\nmid\gcd(h-s,s)$. Hence by Proposition 2.4 we get $v_{p}(d_{\boldsymbol{\lambda}\boldsymbol{\mu}_{s}})\geq t-m$. Thus

[TABLE]

Suppose $m\geq t$. Then

[TABLE]

In both cases we get $v_{L}(d_{\boldsymbol{\lambda}\boldsymbol{\mu}_{s}}c_{\boldsymbol{\lambda}})\geq\tilde{\phi}_{m}(\ell)+h\geq\phi_{j}(\ell)+h$, and hence $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{\rho_{h}(\ell)}}$. Since this holds for $1\leq h\leq n$ we get $\tilde{f}\sim_{\ell}f$.

We now prove the general case. Since $\tilde{f}$ is the minimum polynomial of $\sigma(\tilde{\pi}_{L})$ over $K$ we may assume without loss of generality that $\tilde{\pi}_{L}\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{\ell+1}}$. By repeated application of the special case above we get a sequence $\pi_{L}^{(0)}=\pi_{L},\pi_{L}^{(1)},\pi_{L}^{(2)},\ldots$ of uniformizers for $L$ with minimum polynomials $f^{(0)}=f,f^{(1)},f^{(2)},\ldots$ such that for all $i\geq 0$ we have $\pi_{L}^{(i)}\equiv\tilde{\pi}_{L}\pmod{\mathcal{M}_{L}^{\ell+i+1}}$ and $f^{(i+1)}\sim_{\ell+i}f^{(i)}$. It follows that $f^{(i+1)}\sim_{\ell}f^{(i)}$, and hence that $f^{(i)}\sim_{\ell}f$ for all $i\geq 0$. Since the sequence $(f^{(i)})$ converges coefficientwise to $\tilde{f}$ it follows that $\tilde{f}\sim_{\ell}f$. $\square$

Remark 4.4

It follows from Theorem 4.3 that if $\sigma(\tilde{\pi}_{L})\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{\ell+1}}$ for some $\sigma\in\mathrm{Aut}_{K}(L)$ then $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{\rho_{h}(\ell)}}$ for $1\leq h\leq n$. Define functions $\kappa_{h}:\mathbb{N}\rightarrow\mathbb{N}$ by

[TABLE]

Krasner [7, p. 157] showed that $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{\kappa_{h}(\ell)}}$. Since $\kappa_{h}(\ell)\leq\rho_{h}(\ell)$ Krasner’s congruences are in general weaker than the congruences that follow from Theorem 4.3. However, if $\ell$ is greater than or equal to the largest lower ramification break of $L/K$ then $\phi_{j}(\ell)=\phi_{\nu}(\ell)$ for $0\leq j\leq\nu$. Therefore Theorem 4.3 does not improve on [7] in 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 $n=up^{\nu}$. For $0\leq m\leq\nu$ write the $m$th index of inseparability of $L/K$ in the form $i_{m}=A_{m}n-b_{m}$ with $1\leq b_{m}\leq n$. Let $\pi_{L}$, $\tilde{\pi}_{L}$ be uniformizers for $L$ such that there are $\ell\geq 1$, $r\in\mathcal{O}_{K}$, and $\sigma\in\mathrm{Aut}_{K}(L)$ with $\sigma(\tilde{\pi}_{L})\equiv\pi_{L}+r\pi_{L}^{\ell+1}\pmod{\mathcal{M}_{L}^{\ell+2}}$. Let $0\leq j\leq\nu$ satisfy $\overline{v}_{p}(\phi_{j}(\ell))=j$, and let $h$ be the unique integer such that $1\leq h\leq n$ and $n$ divides $\phi_{j}(\ell)+h$. Set $k=(\phi_{j}(\ell)+h)/n$ and $h_{0}=h/p^{j}$. Then

[TABLE]

where

[TABLE]

*Proof: *We first prove that the theorem holds for $\hat{\pi}_{L}=\pi_{L}+r\pi_{L}^{\ell+1}$. Let

[TABLE]

be the minimum polynomial for $\hat{\pi}_{L}$ over $K$. Let $1\leq s\leq h$ and let $\boldsymbol{\lambda}$ be a partition of $\ell s+h$ whose parts are at most $n$. Choose $q$ to minimize $\overline{v}_{p}(\lambda_{q})$ and set $t=\overline{v}_{p}(\lambda_{q})$. Recall that $\boldsymbol{\mu}_{s}$ is the partition of $\ell s+h$ consisting of $h-s$ copies of 1 and $s$ copies of $\ell+1$. Since $\overline{v}_{p}(h)=\overline{v}_{p}(\phi_{j}(\ell))=j$ it follows from the proof of Theorem 4.3 that $v_{K}(d_{\boldsymbol{\lambda}\boldsymbol{\mu}_{s}}c_{\boldsymbol{\lambda}})\geq k$. Suppose $v_{K}(d_{\boldsymbol{\lambda}\boldsymbol{\mu}_{s}}c_{\boldsymbol{\lambda}})=k$. Then the inequalities in the proof of Theorem 4.3 must be equalities. Hence there is $0\leq m\leq j$ such that $s=p^{m}$, $v_{L}(c_{\boldsymbol{\lambda}})=i_{t}^{\pi_{L}}+\ell p^{m}+h$, and $\phi_{j}(\ell)=\tilde{\phi}_{m}(\ell)$. In particular, we have $m\in S_{j}$.

Let $w_{m}=\ell p^{m}+h$ and let $\boldsymbol{\kappa}_{m}$ be the partition of $w_{m}$ consisting of $k-A_{m}$ copies of $n$ and 1 copy of $b_{m}$. By Proposition 3.2 we see that $\boldsymbol{\lambda}$ has at most one element not equal to $n$. Since $\boldsymbol{\lambda}$ is a partition of $w_{m}$, and

[TABLE]

it follows that $\boldsymbol{\lambda}=\boldsymbol{\kappa}_{m}$. Hence $c_{\boldsymbol{\lambda}}=c_{\boldsymbol{\kappa}_{m}}=c_{n}^{k-A_{m}}c_{b_{m}}$ and $\overline{v}_{p}(b_{m})=\overline{v}_{p}(\lambda_{q})=t$. Using equation (4.1) and Proposition 4.2 we get

[TABLE]

Let $m\in S_{j}$. Since

[TABLE]

and $m\leq j$ we get $m\leq\overline{v}_{p}(i_{m})=\overline{v}_{p}(b_{m})$. Hence $b_{m}^{\prime}=b_{m}/p^{m}$ is an integer. Let $\boldsymbol{\kappa}_{m}^{\prime}$ be the partition of

[TABLE]

consisting of $k-A_{m}$ copies of $up^{\nu-m}$ and 1 copy of $b_{m}^{\prime}$. Let $\boldsymbol{\mu}_{p^{m}}^{\prime}$ be the partition of $w_{m}^{\prime}$ consisting of $h_{0}p^{j-m}-1$ copies of 1 and 1 copy of $\ell+1$. Since $h\leq n$ we have $up^{\nu-m}>h_{0}p^{j-m}-1$. Hence if $b_{m}^{\prime}\not=up^{\nu-m}$ then we can compute $d_{\boldsymbol{\kappa}_{m}^{\prime}\boldsymbol{\mu}_{p^{m}}^{\prime}}$ using Proposition 2.3.

Suppose $b_{m}<h$. Then $h_{0}p^{j-m}-1\geq b_{m}^{\prime}$, so by Proposition 2.3 we get

[TABLE]

Suppose $h\leq b_{m}<n$. Then $h_{0}p^{j-m}-1<b_{m}^{\prime}$, so by Proposition 2.3 we get

[TABLE]

Suppose $b_{m}=n$, so that $b_{m}^{\prime}=up^{\nu-m}$. Since $up^{\nu-m}>h_{0}p^{j-m}-1$, the only cycle digraph which admits a $(\boldsymbol{\kappa}_{m}^{\prime},\boldsymbol{\mu}_{p^{m}}^{\prime})$-tiling consists of a single cycle $\Gamma$ of length $w_{m}^{\prime}$. By Proposition 2.2(a) we get $\eta_{\boldsymbol{\kappa}_{m}^{\prime}\boldsymbol{\mu}_{p^{m}}^{\prime}}(\Gamma)=up^{\nu-m}$. It then follows from (2.1) that

[TABLE]

Hence in all three cases we have $d_{\boldsymbol{\kappa}_{m}^{\prime}\boldsymbol{\mu}_{p^{m}}^{\prime}}=g_{m}$.

Since $m\leq t\leq\nu$ it follows from (3.2) and (3.1) that

[TABLE]

Since $p^{t}\mid b_{m}$ we have $p^{t-m}\mid b_{m}^{\prime}$. Therefore by Proposition 2.5 we get

[TABLE]

Using (4.3) we see that

[TABLE]

Therefore the theorem holds when $\tilde{\pi}_{L}=\hat{\pi}_{L}$.

We now prove the theorem in the general case. We may assume that

[TABLE]

It follows that $\tilde{\pi}_{L}\equiv\hat{\pi}_{L}\pmod{\mathcal{M}_{L}^{\ell+2}}$, so by Theorem 4.3 we get $\tilde{c}_{h}\equiv\hat{c}_{h}\pmod{\mathcal{M}_{K}^{\rho_{h}(\ell+1)}}$. Since $(\phi_{j}(\ell)+h)/n=k$ and $\phi_{j}(\ell+1)>\phi_{j}(\ell)$ this implies $\tilde{c}_{h}\equiv\hat{c}_{h}\pmod{\mathcal{M}_{K}^{k+1}}$. Hence the theorem holds for $\tilde{\pi}_{L}$. $\square$

Remark 4.6

Suppose $\overline{v}_{p}(\phi_{j}(\ell))=j^{\prime}\leq j$. Then $\phi_{j}(\ell)=\phi_{j^{\prime}}(\ell)$. In particular, $\phi_{\nu}(\ell)=\phi_{j^{\prime}}(\ell)$ with $j^{\prime}=\overline{v}_{p}(\phi_{\nu}(\ell))$. Hence if $1\leq h\leq n$ and $n$ divides $\phi_{\nu}(\ell)+h$ then Theorem 4.5 gives a congruence for $\tilde{c}_{h}$ modulo $\mathcal{M}_{K}^{k+1}$, where $k=(\phi_{\nu}(\ell)+h)/n$. This is the congruence obtained by Krasner [7, p. 157]. If $\ell$ is greater than or equal to the largest lower ramification break of $L/K$ then $\phi_{j}(\ell)=\phi_{\nu}(\ell)$ for $0\leq j\leq\nu$. Therefore Theorem 4.5 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 $\mathbb{Q}_{3}$ such that $v_{K}(3)\geq 2$. Let

[TABLE]

be an Eisenstein polynomial over $K$ such that $v_{K}(c_{2})=v_{K}(c_{6})=2$, $v_{K}(c_{h})\geq 2$ for $h\in\{1,3\}$, and $v_{K}(c_{h})\geq 3$ for $h\in\{4,5,7,8\}$. Let $\pi_{L}$ be a root of $f(X)$. Then $L=K(\pi_{L})$ is a totally ramified extension of $K$ of degree 9, so we have $u=1$, $\nu=2$. It follows from our assumptions about the valuations of the coefficients of $f(X)$ that the indices of inseparability of $L/K$ are $i_{0}=16$, $i_{1}=12$, and $i_{2}=0$. Therefore $A_{0}=2$, $A_{1}=2$, $A_{2}=1$, and $b_{0}=2$, $b_{1}=6$, $b_{2}=9$. We get the following values for $\tilde{\phi}_{j}(\ell)$ and $\phi_{j}(\ell)$:

[TABLE]

Now let $\tilde{\pi}_{L}$ be another uniformizer for $L$, with minimum polynomial

[TABLE]

Suppose $\tilde{\pi}_{L}\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{2}}$. Then by Theorem 4.3 we get $\tilde{f}\sim_{1}f$. Using the table above we find that

[TABLE]

This is an improvement on [7], which gives $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{2}}$ for $1\leq h\leq 9$. If $\tilde{\pi}_{L}\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{3}}$ we get $\tilde{f}\sim_{2}f$, and hence $\tilde{c}_{h}\equiv c_{h}\pmod{\mathcal{M}_{K}^{3}}$ for $1\leq h\leq 9$. If $\tilde{\pi}_{L}\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{4}}$ we get $\tilde{f}\sim_{3}f$, and hence

[TABLE]

Since the largest lower ramification break of $L/K$ is 2, the congruences we get for $\ell\geq 2$ are the same as those in [7].

Suppose $\tilde{\pi}_{L}\equiv\pi_{L}+r\pi_{L}^{2}\pmod{\mathcal{M}_{L}^{3}}$, with $r\in\mathcal{O}_{K}$. By the table above we get $\overline{v}_{3}(\phi_{0}(1))=0$, $\overline{v}_{3}(\phi_{1}(1))=1$, $\overline{v}_{3}(\phi_{2}(1))=2$ and $S_{0}=\{0\}$, $S_{1}=\{1\}$, $S_{2}=\{2\}$. The corresponding values of $h$ are 1, 3, 9, and we have $h_{0}=1$, $k=2$ in all three cases. By applying Theorem 4.5 with $\ell=1$, $j=0,1,2$ we get the following congruences:

[TABLE]

Only the congruence for $\tilde{c}_{9}$ follows from [7].

Suppose $\tilde{\pi}_{L}\equiv\pi_{L}+r\pi_{L}^{3}\pmod{\mathcal{M}_{L}^{4}}$. Then $\overline{v}_{3}(\phi_{2}(2))=2$ and $S_{2}=\{0,1,2\}$, which gives $h=9$, $h_{0}=1$, and $k=3$. By applying Theorem 4.5 with $\ell=2$, $j=2$ we get the following congruence:

[TABLE]

Suppose $\tilde{\pi}_{L}\equiv\pi_{L}+r\pi_{L}^{4}\pmod{\mathcal{M}_{L}^{5}}$. Then $\overline{v}_{3}(\phi_{0}(3))=0$ and $S_{0}=\{0\}$, so we get $h=8$, $h_{0}=8$, and $k=3$. By applying Theorem 4.5 with $\ell=3$, $j=0$ we get the following congruence:

[TABLE]

Again, since the largest lower ramification break of $L/K$ is 2, the congruences we get for $\ell\geq 2$ are the same as those in [7]. $\square$

One might hope to prove the following converse to Theorem 4.3: If $\pi_{L}$, $\tilde{\pi}_{L}$ are uniformizers for $L$ whose minimum polynomials satisfy $\tilde{f}\sim_{\ell}f$, then there is $\sigma\in\mathrm{Aut}_{K}(L)$ such that $\sigma(\tilde{\pi}_{L})\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{\ell+1}}$. The example below shows that this is not necessarily the case:

Example 5.2

Let $\pi_{L}$ be a root of the Eisenstein polynomial $f(X)=X^{4}+6X^{2}+4X+2$ over the 2-adic field $\mathbb{Q}_{2}$. Then $L=\mathbb{Q}_{2}(\pi_{L})$ is a totally ramified extension of $\mathbb{Q}_{2}$ of degree 4, with indices of inseparability $i_{0}=5$, $i_{1}=2$, and $i_{2}=0$. We get the following values for $\tilde{\phi}_{j}(\ell)$ and $\phi_{j}(\ell)$:

[TABLE]

Set $\tilde{\pi}_{L}=\pi_{L}+\pi_{L}^{2}$, and let the minimum polynomial for $\tilde{\pi}_{L}$ over $\mathbb{Q}_{2}$ be

[TABLE]

By Theorem 4.3 we have $\tilde{f}\sim_{1}f$, and hence

[TABLE]

Theorem 4.5 gives a refinement of the last congruence:

[TABLE]

Using this refinement we get $\tilde{f}\sim_{2}f$.

Using [5] (see also Table 4.2 in [4]) we obtain a list of the degree-4 extensions of $\mathbb{Q}_{2}$. Using the data in this list we find that $L/\mathbb{Q}_{2}$ is not Galois, and the only quadratic subextension of $L/\mathbb{Q}_{2}$ is $M/\mathbb{Q}_{2}$, where $M=\mathbb{Q}_{2}(\sqrt{-1})$. Hence $\mathrm{Aut}_{\mathbb{Q}_{2}}(L)=\mathrm{Gal}(L/M)$. Since the lower ramification breaks of $L/\mathbb{Q}_{2}$ are 1, 3, and the lower ramification break of $M/\mathbb{Q}_{2}$ is 1, the lower ramification break of $L/M$ is 3. Hence if $\sigma\in\mathrm{Aut}_{\mathbb{Q}_{2}}(L)$ then $\sigma(\tilde{\pi}_{L})\equiv\tilde{\pi}_{L}\pmod{\mathcal{M}_{L}^{4}}$. Since $\tilde{\pi}_{L}=\pi_{L}+\pi_{L}^{2}$ we get $\sigma(\tilde{\pi}_{L})\not\equiv\pi_{L}\pmod{\mathcal{M}_{L}^{3}}$. $\square$