Invariants of algebraic elements over henselian fields

TL;DR

This paper explores how Okutsu sequences and MacLane chains can be used to compute invariants of algebraic elements over henselian valued fields, advancing understanding of valuation theory.

Contribution

It introduces a novel approach linking Okutsu sequences with MacLane chains to compute algebraic invariants over henselian fields.

Findings

Established connections between Okutsu sequences and MacLane chains.

Provided methods for computing invariants of algebraic elements.

Enhanced valuation theory techniques for henselian fields.

Abstract

Let (K,v) be a henselian valued field. In this paper, we use Okutsu sequences for monic, irreducible polynomials in K[x], and their relationship with MacLane chains of inductive valuations on K[x], to obtain some results on the computation of invariants of algebraic elements over K.

Equations

$$W(\theta)=\left\{\dfrac{v(g(\theta))}{\deg(g)}\;\Big{|}\;g\in K[x]\mbox{ monic},\ 0<\deg(g)<n\right\}\subset\mathbb{Q}\Gamma.$$

$$w(F)=w(\theta )=\mathrm{Max}\left(W(\theta )\right).$$

$$\Delta (\theta )=\{v(\theta -\alpha )\mid \alpha \in \overline{K},{\mathrm{deg}}_K(\alpha )<\mathrm{deg}(\theta )\}.$$

$$\delta (F)=\delta (\theta )=\mathrm{Max}\left(\Delta (\theta )\right).$$

$$\omega (F)=\omega (\theta )=\mathrm{Max}\left(\Omega (\theta )\right),\Omega (\theta )=\{v(\theta -{\theta }^{\prime })\mid {\theta }^{\prime }\in Z(F),{\theta }^{\prime }\ne \theta \},$$

$${\mathcal{P}}_{\alpha }=\{g\in K[x]\mid \mu (g)\ge \alpha \}\supset {\mathcal{P}}_{\alpha }^{+}=\{g\in K[x]\mid \mu (g)>\alpha \}.$$

$${\mathcal{G}}_{\mu }:={\mathrm{gr}}_{\mu }(K[x])={⨁}_{\alpha \in {\Gamma }_{\mu }}{\mathcal{P}}_{\alpha }/{\mathcal{P}}_{\alpha }^{+}.$$

$$k\text{lhook}\text{joinrel}⟶{\Delta }_{\mu }\text{lhook}\text{joinrel}⟶k_{\mu }.$$

$$H_{\mu }(g)=g+{\mathcal{P}}_{\mu (g)}^{+}\in {\mathcal{P}}_{\mu (g)}/{\mathcal{P}}_{\mu (g)}^{+},\text{mbox}ifg\ne 0.$$

$$\mu (f)/\mathrm{deg}(f)\le w(\mu ):=\mu (\varphi )/\mathrm{deg}(\varphi ),$$

$$f=\sum _{0\le s}{a}_s{\varphi }^s,a_s\in K[x],\mathrm{deg}(a_s)<\mathrm{deg}(\varphi ).$$

$${\mu }^{\prime }:K[x]\to {\Gamma }^{\prime }\cup \left\{\infty \right\},{\mu }^{\prime }(f)=\mathrm{Min}\left\{\mu (a_s)+s\gamma \mid 0\le s\right\}.$$

$$\mu (\varphi )<\gamma ={\mu }^{\prime }(\varphi ),\mu (f)\le {\mu }^{\prime }(f),\forall f\in K[x].$$

$$v_{\varphi }:K[x]\text{relbar}\text{joinrel}\twoheadrightarrow K_{\varphi }\stackrel{v}{⟶}\mathbb{Q}\Gamma \cup \{\infty \}.$$

$${\mu }_{-\infty }\stackrel{{\varphi }_0,{\gamma }_0}{⟶}{\mu }_0\stackrel{{\varphi }_1,{\gamma }_1}{⟶}{\mu }_1\stackrel{{\varphi }_2,{\gamma }_2}{⟶}\cdots \stackrel{{\varphi }_{r-1},{\gamma }_{r-1}}{⟶}{\mu }_{r-1}\stackrel{{\varphi }_r,{\gamma }_r}{⟶}{\mu }_r=\mu ,$$

$${\mu }_0\left({\sum }_{0\le s}{a}_s{\varphi }_0^s\right)=\mathrm{Min}\left\{v(a_s)+s{\gamma }_0\mid 0\le s\right\}.$$

$$1=m_0\mid \cdots \mid m_r,m_0<\cdots <m_r.$$

$${\lambda }_0={\gamma }_0=\mu ({\varphi }_0),{\lambda }_i={\mu }_i({\varphi }_i)-{\mu }_{i-1}({\varphi }_i)>0,0<i\le r.$$

$$e_0=1,e_i=\left({\Gamma }_{{\mu }_i}:{\Gamma }_{{\mu }_{i-1}}\right),0<i\le r.$$

$$k\simeq k_{{\varphi }_0}⟶k_{{\varphi }_1}⟶\cdots ⟶k_{{\varphi }_r}⟶k_{\varphi }.$$

$$e({\varphi }_i):=e\left(K_{{\varphi }_i}/K\right)=\left({\Gamma }_{{\mu }_{i-1}}:\Gamma \right)=e_0\cdots e_{i-1}.$$

$$\frac{{\gamma }_i}{m_i}=\frac{{\mu }_i({\varphi }_i)}{m_i}=w({\mu }_i)=\frac{{\mu }_i({\varphi }_{i+1})}{m_{i+1}}=\frac{{\gamma }_{i+1}-{\lambda }_{i+1}}{m_{i+1}}.$$

$$\frac{{\gamma }_i}{m_i}=\frac{{\lambda }_0}{m_0}+\cdots +\frac{{\lambda }_i}{m_i},0\le i\le r.$$

$${\mu }_i:K[x]⟶\mathbb{Q}\Gamma \cup \{\infty \},{\sum }_{0\le s}{a}_s{\varphi }_i^s⟼\mathrm{Min}\{v(a_s(\theta ))+s{\gamma }_i\mid 0\le s\},$$

$${\mu }_{-\infty }\stackrel{{\varphi }_0,{\gamma }_0}{⟶}{\mu }_0\stackrel{{\varphi }_1,{\gamma }_1}{⟶}{\mu }_1\stackrel{{\varphi }_2,{\gamma }_2}{⟶}\cdots \stackrel{{\varphi }_{r-1},{\gamma }_{r-1}}{⟶}{\mu }_{r-1}\stackrel{{\varphi }_r,{\gamma }_r}{⟶}{\mu }_r.$$

$$w(F)=\frac{{\gamma }_r}{m_r}=\frac{{\lambda }_0}{m_0}+\cdots +\frac{{\lambda }_r}{m_r}.$$

$${\mathrm{deg}}_K({\beta }_{\text{mbox}sep})={\mathrm{deg}}_K(\beta ),v(\beta -{\beta }_{\text{mbox}sep})>\rho .$$

$$v(\pi )>{\mathrm{deg}}_K(\beta )\rho -v(\beta ).$$

$${\sum }_{\alpha \in Z(g_{\text{mbox}sep})}v(\beta -\alpha )=v\left(g_{\text{mbox}sep}(\beta )\right)=v(\pi \beta )=v(\pi )+v(\beta )>{\mathrm{deg}}_K(\beta )\rho .$$

$$v(\theta -\beta )\ge \delta ⟹\frac{v(g(\theta ))}{\mathrm{deg}(g)}\ge \frac{v(\varphi (\theta ))}{\mathrm{deg}(\varphi )}.$$

Full text

Partially supported by grant 204224/2014-4 from CNPq

Introduction

Let $(K,v)$ be a henselian field and denote still by $v$ the canonical extension of this valuation to a fixed algebraic closure $\overline{K}$ of $K$. Let $\Gamma=v(K^{*})$ be the value group of $v$ over $K$, and denote by $\mathbb{Q}\Gamma=\Gamma\otimes\mathbb{Q}$ the value group of $v$ over $\overline{K}$.

Let $F\in K[x]$ be a monic, irreducible polynomial of degree $n>1$. Denote by $Z(F)\subset\overline{K}$ the set of rootos of $F$, and take $\theta\in Z(F)$. Consider the following set

[TABLE]

If this set contains a maximal element, we define the weight of $\theta$ (or $F$) as:

[TABLE]

A pair $\phi,F$ is a distinguished pair of polynomials if $\phi\in K[x]$ is a monic polynomial of minimal degree satisfying $v(\phi(\theta))/\deg(\phi)=w(F)$.

An Okutsu frame of $F$ is a list $[\phi_{0},\phi_{1},\dots,\phi_{r}]$ of monic, irreducible polynomials in $K[x]$, such that $\deg(\phi_{0})=1$ and $\phi_{i},\phi_{i+1}$ is a distinguished pair for all $0\leq i\leq r$, where we agree that $\phi_{r+1}=F$.

If $F$ admits an Okutsu frame, we say that $F$ is an Okutsu polynomial, and the index $r\geq 0$ is called the Okutsu depth of $F$.

A monic, irreducible polynomial $F\in K[x]$ is an Okutsu polynomial if and only if it is defectless; that is, $\deg(F)=e(K(\theta)/K)f(K(\theta)/K)$. As a consequence, there is a tight link between defectless polynomials and key polynomials for inductive valuations on $K[x]$ [6, Sec. 10.4]. In particular, we may attach to any defectless polynomial $F$ a bunch of arithmetic invariants, which can be read in any optimal MacLane chain of a certain inductive valuation on $K[x]$ canonically associated with $F$ (cf. section 1.2).

In Corollary 1.7 we give an explicit formula for the weight of a defectless polynomial in terms of these invariants.

On the other hand, the value $v(g(\theta))/\deg(g)$ is the average of the values $v(\theta-\beta)$ for $\beta$ running on the roots of $g$ in $\overline{K}$. Let us focus our attention on these latter values, and consider the set

[TABLE]

If $F$ is defectless, this set contains a maximal value which is called the main invariant of $\theta$ (or $F$), and is denoted

[TABLE]

In this paper, we use the connection between defectless polynomials and MacLane chains of inductive valuations to prove that, if $\theta$ is quasi-tame over $K$, then the main invariant coincides with Krasner’s constant:

[TABLE]

where we consider $\Omega(\theta)$ as a multiset of cardinality $n-1$.

Also, we find explicit formulas for all values in the multiset $\Omega(\theta)$, and for their multiplicities, in terms of the discrete invariants supported by any optimal MacLane chain corresponding to $F$.

Most of these results can be found in the literature as the combined contribution of several authors. Special mention deserve Aghigh-Khanduja [1, 2], Brown-Merzel [3, 4] and Singh-Khanduja [10].

Our approach yields more direct proofs and a new arithmetic interpretation of the values of the multiset $\Omega(\theta)$ in the case that $\theta$ is quasi-tame.

1. Inductive valuations and Okutsu frames - A short background

We keep dealing with our henselian valued field $(K,v)$.

In this section, we review some basic facts on valuations on $K[x]$, mainly extracted from [11] and [6].

1.1. Key polynomials and augmentation of valuations

Let $\mu$ be a valuation on $K(x)$ extending $v$.

Let $\Gamma_{\mu}=\mu\left(K(x)^{*}\right)$ be the value group, and $k_{\mu}$ the residue class field of $\mu$.

For any $\alpha\in\Gamma_{\mu}$, consider the abelian groups:

[TABLE]

The graded algebra of $\mu$ over $K[x]$ is the integral domain:

[TABLE]

Let $\Delta_{\mu}=\mathcal{P}_{0}/\mathcal{P}_{0}^{+}\subset\mathcal{G}_{\mu}$ be the subring of homogeneous elements of degree zero. There are canonical injective ring homomorphisms:

[TABLE]

In particular, $\Delta_{\mu}$ and $\mathcal{G}_{\mu}$ are equipped with a canonical structure of $k$-algebra.

There is a natural map $H_{\mu}\colon K[x]\to\mathcal{G}_{\mu}$, given by $H_{\mu}(0)=0$ and

[TABLE]

Definition 1.1**.**

A key polynomial for $\mu$ is a monic polynomial $\phi\in K[x]$ such that

• •

$H_{\mu}(\phi)$* is a prime element in $\mathcal{G}_{\mu}$.*

• •

For all $f\in K[x]$ with $\deg(f)<\deg(\phi)$, the prime element $H_{\mu}(\phi)$ does not divide $H_{\mu}(f)$ in $\mathcal{G}_{\mu}$.

A key polynomial is necessarily irreducible in $K[x]$.

Theorem 1.2**.**

[7, Thm. 3.9]** Let $\phi\in K[x]$ be a key polynomial for $\mu$. For any monic non-constant $f\in K[x]$ we have

[TABLE]

and equality holds if $f$ is a key polynomial for $\mu$.

We say that $w(\mu)$ is the weight of $\mu$.

Consider an order-preserving embedding $\iota\colon\Gamma_{\mu}\hookrightarrow\Gamma^{\prime}$ of ordered abelian groups. Take $\phi\in K[x]$ a key polynomial for $\mu$, and $\gamma\in\Gamma^{\prime}$ any element such that $\mu(\phi)<\gamma$.

For any $f\in K[x]$, consider its canonical $\phi$-expansion

[TABLE]

Then, the following mapping is a valuation on $K[x]$:

[TABLE]

We say that $\mu^{\prime}=[\mu;\phi,\gamma]$ is an augmented valuation of $\mu$. It satisfies:

[TABLE]

Hence, we have a natural homomorphism of graded algebras $\mathcal{G}_{\mu}\to\mathcal{G}_{\mu^{\prime}}$.

Lemma 1.3**.**

Let $\mu^{\prime}=[\mu;\phi,\gamma]$. Then, $\phi$ is a key polynomial for $\mu^{\prime}$.

Moreover, let $K_{\phi}=K[x]/{\phi K[x]}$ and consider the semivaluation:

[TABLE]

Then, the group of values $\Gamma_{v_{\phi}}$ is equal to $\Gamma_{\mu}$.

1.2. Inductive valuations

A valuation $\mu$ on $K[x]$, extending $v$, is said to be inductive if it is attained after a finite number of augmentation steps:

[TABLE]

with $\gamma_{0},\dots,\gamma_{r}\in\mathbb{Q}\Gamma$, and intermediate valuations $\mu_{i}=[\mu_{i-1};\phi_{i},\gamma_{i}]$, for $0<i\leq r$.

The valuation $\mu_{-\infty}$ is an incommensurable extension of $v$ to $K[x]$, playing the role of absolute minimal extension. Since we are not going to use it, let us simply say that its key polynomial $\phi_{0}\in K[x]$ has degree one, and its augmentation $\mu_{0}$ is defined as:

[TABLE]

Denote $m_{i}=\deg(\phi_{i})$ for $0\leq i\leq r$.

An optimal MacLane chain of $\mu$ is any chain (3) of augmentations satisfying

[TABLE]

All inductive valuations admit optimal MacLane chains. These chains are not unique, but they support many intrinsic data of $\mu$:

• •

The intermediate valuations $\mu_{0},\dots,\mu_{r-1}$.

• •

The degrees $m_{0},\dots,m_{r}$ of the key polynomials.

• •

The slopes $\gamma_{0},\cdots,\gamma_{r}$, which satisfy $\gamma_{i}=\mu_{i}(\phi_{i})=\mu(\phi_{i})$ for all $0\leq i\leq r$.

• •

The secondary slopes $\lambda_{0},\cdots,\lambda_{r}$, defined as

[TABLE]

• •

The relative ramification indices $e_{0},\dots,e_{r-1}$, defined as

[TABLE]

For any key polynomial $\phi$ for $\mu$, there is a tower of fields:

[TABLE]

The identification $\Gamma_{\mu_{i-1}}=\Gamma_{v_{\phi_{i}}}$, given in Lemma 1.3, allows a computation of the ramification index of the extension $K_{\phi_{i}}/K$ in terms of these data:

[TABLE]

Finally, Lemma 1.3 shows that each intermediate valuation $\mu_{i}$ admits $\phi_{i}$ and $\phi_{i+1}$ as key polynomials. By Theorem 1.2,

[TABLE]

This relates the main and secondary slopes by an explicit formula:

[TABLE]

1.3. Okutsu frames

Let us quote some fundamental results of [6].

Theorem 1.4**.**

Consider an optimal MacLane chain of an inductive valuation $\mu$ as in (3). Let $F$ be a key polynomial for $\mu$. Then, $F$ is an Okutsu polynomial, and

1. (1)

If $\deg(F)>\deg(\phi_{r})$, then $[\phi_{0},\dots,\phi_{r}]$ is an Okutsu frame of $F$. 2. (2)

If $\deg(F)=\deg(\phi_{r})$, then $[\phi_{0},\dots,\phi_{r-1}]$ is an Okutsu frame of $F$.

Moreover, $w(F)=w(\mu)$ if $\deg(F)>\deg(\phi_{r})$.

Theorem 1.5**.**

Let $F\in K[x]$ be an Okutsu polynomial, and let $\theta\in\overline{K}$ be a root of $F$. Let $[\phi_{0},\dots,\phi_{r}]$ be an Okutsu frame of $\phi_{r+1}=F$. For all $0\leq i\leq r$, denote $\gamma_{i}=v(\phi_{i}(\theta))$ and consider the mapping

[TABLE]

where $\deg(a_{s})<\deg(\phi_{i})$ for all $s\geq 0$.

Then, $\mu_{i}$ is a valuation, $\phi_{i+1}$ is a key polynomial for $\mu_{i}$, and $\mu_{r}$ admits an optimal MacLane chain

[TABLE]

The next result follows from Theorems 1.4, 1.5 and from [12].

Theorem 1.6**.**

Let $F$ be a monic irreducible polynomial in $K[x]$. The following conditions are equivalent:

1. (1)

$F$* is the key polynomial of an inductive valuation.* 2. (2)

$F$* is an Okutsu polynomial.* 3. (3)

$F$* is defectless.*

Corollary 1.7**.**

Let $F\in K[x]$ be a monic, irreducible defectless polynomial. Then, if $r$ is the Okutsu depth of $F$, we have

[TABLE]

where $\gamma_{i}$, $m_{i}$, $\lambda_{i}$ are the intrinsic data of any optimal MacLane chain of the inductive valuation attached to $F$ in Theorem 1.6.

2. Complete distinguished chains of defectless algebraic elements

We keep dealing with a henselian valued field $(K,v)$.

In section 2.1 we connect distinguished pairs of polynomials with distinguished pairs of algebraic elements.

Distinguished pairs of algebraic elements and complete distinguished chains were introduced by N. Popescu-A. Zaharescu in 1995, for $K$ a complete, discrete, rank-one valued field [9]. However, these objects are equivalent to some sequences of algebraic elements studied by Okutsu in 1982, also in the complete and discrete rank-one case [8]. In section 2.2 we show the equivalence between the two concepts, for arbitrary henselian fields.

2.1. Distinguished pairs of algebraic elements

Let $F\in K[x]$ be a monic, irreducible polynomial of degree $n>1$, and let $\theta\in\overline{K}$ be a rot of $F$.

In this section, we prove that the set $W(\theta)$ in (1) contains a maximal value if and only if the set $\Delta(\theta)$ in (2) contains a maximal value.

Lemma 2.1**.**

For any $\beta\in\overline{K}$ inseparable over $K$, and any $\rho\in\mathbb{Q}\Gamma$, there exists $\beta_{\mbox{\tiny sep}}\in\overline{K}$ separable over $K$ such that

[TABLE]

Proof.

Let $g\in K[x]$ be the minimal polynomial of $\beta$ over $K$. We have $g^{\prime}=0$.

Consider the polynomial $g_{\mbox{\tiny sep}}=g+\pi x\in K[x]$, where $\pi\in K^{*}$ satisfies

[TABLE]

Since $g_{\mbox{\tiny sep}}^{\prime}=\pi\neq 0$, this polynomial is separable. On the other hand,

[TABLE]

Hence, there exists $\alpha\in\operatorname{Z}(g_{\mbox{\tiny sep}})$ such that $v(\beta-\alpha)>\rho$. We may take $\beta_{\mbox{\tiny sep}}=\alpha$. ∎

Definition 2.2**.**

Let $\alpha\in\overline{K}$ with $\deg_{K}(\alpha)<n$.

We say that $\alpha,\,\theta$ is a distinguished pair if the two following conditions are satisfied:

1. (1)

$v(\theta-\alpha)=\delta(\theta)$. 2. (2)

$\beta\in\overline{K},\quad\deg_{K}(\beta)<\deg_{K}(\alpha)\ \,\Longrightarrow\,\ v(\theta-\beta)<\delta(\theta)$.

Theorem 2.3**.**

**

1. (1)

Suppose that $\phi,\,F$ is a distinguished pair of polynomials. If $\alpha\in\operatorname{Z}(\phi)$ has $v(\theta-\alpha)=\operatorname{Max}\{v(\theta-\alpha^{\prime})\mid\alpha^{\prime}\in\operatorname{Z}(\phi)\}$, then $\alpha,\theta$ is a distinguished pair. 2. (2)

Suppose that $\alpha,\,\theta$ is a distinguished pair. If $\phi\in K[x]$ is the minimal polynomial of $\alpha$ over $K$, then $\phi,\,F$ is a distinguished pair of polynomials.

Proof.

Let us first check (1). Suppose that $\phi,\,F$ is a distinguished pair of polynomials. Let $\delta=v(\theta-\alpha)=\operatorname{Max}\{v(\theta-\alpha^{\prime})\mid\alpha^{\prime}\in\operatorname{Z}(\phi)\}$.

Consider any $\beta\in\overline{K}$ with $\deg_{K}(\beta)<n$. We want to show:

1. (i) 

$v(\theta-\beta)\leq\delta$. 2. (ii)

$v(\theta-\beta)=\delta\ \,\Longrightarrow\,\ \deg_{K}(\beta)\geq\deg_{K}(\alpha)$.

Let $g\in K[x]$ be the minimal polynomial of $\beta$ over $K$. We may assume that $v(\theta-\beta)=\operatorname{Max}\{v(\theta-\beta^{\prime})\mid\beta^{\prime}\in\operatorname{Z}(g)\}$. By Lemma 2.1, we may assume too, that $\theta$, $\alpha$ and $\beta$ are separable over $K$. Consider a finite Galois extension $M/K$ containing $\theta$, $\alpha$ and $\beta$, and denote $G=\operatorname{Gal}(M/K)$. We claim that

[TABLE]

In fact, assume that $v(\theta-\beta)\geq\delta$. Then, for any $\sigma\in G$ we get:

[TABLE]

because $v(\theta-\sigma(\alpha))\leq\delta$, while $v(\alpha-\theta),\,v(\theta-\beta)\geq\delta$. Therefore,

[TABLE]

This proves the claimed implication (7).

Now, if we had $v(\theta-\beta)>\delta$, then at least for the automorphism $\sigma=1$ we would have $v(\theta-\sigma(\beta))>\delta=v(\theta-\sigma(\alpha))$, leading to a strict inequality in (9). This would contradict the fact that $\phi,F$ is a distinguished pair. This argument proves (i).

On the other hand, the equality $v(\theta-\beta)=\delta$ is incompatible with a strict inequality in (9). In fact, suppose that for some $\sigma\in G$ we had

[TABLE]

Then, the inequality in (8) becomes an equality, and this contradicts our assumptions:

[TABLE]

Thus, if $v(\theta-\beta)=\delta$, we must have an equality in (9). Since $\phi,F$ is a distinguished pair, this implies $\deg(g)\geq\deg(\phi)$. This proves (ii).

Let us now prove (2). Suppose that $\alpha,\,\theta$ is a distinguished pair, and keep the notation  $\delta=v(\theta-\alpha)=\operatorname{Max}\{v(\theta-\alpha^{\prime})\mid\alpha^{\prime}\in\operatorname{Z}(\phi)\}$.

Let $g\in K[x]$ be a monic polynomial with $\deg(g)<n$. We want to show:

1. (i) 

$v(g(\theta))/\deg(g)\,\leq\,v(\phi(\theta))/\deg(\phi)$. 2. (ii)

$v(g(\theta))/\deg(g)=v(\phi(\theta))/\deg(\phi)\ \,\Longrightarrow\,\ \deg(g)\geq\deg(\phi)$.

By Lemma 2.4 below, we may assume that $g$ is irreducible and separable. Also, by Lemma 2.1, we may assume that $\alpha$ and $\theta$ are separable too. Let $M/K$ be a finite Galois extension containing $\theta$, $\alpha$ and $\beta$, and denote $G=\operatorname{Gal}(M/K)$.

Take $\beta\in\operatorname{Z}(g)$ such that $v(\theta-\beta)=\operatorname{Max}\{v(\theta-\beta^{\prime})\mid\beta^{\prime}\in\operatorname{Z}(g)\}$, then for all $\sigma\in G$,

[TABLE]

Now, we claim that

[TABLE]

In fact, if $v\left(\theta-\sigma(\alpha)\right)=\delta$, then our assumption is a consequence of the fact that $\alpha,\,\theta$ is a distinguished pair. If $v\left(\theta-\sigma(\alpha)\right)<\delta$, then the claim follows from (10). Indeed,

[TABLE]

because $v\left(\theta-\sigma(\alpha)\right)<\delta=v\left(\sigma(\alpha)-\sigma(\theta)\right)$.

From the claim it follows that

[TABLE]

Also, if equality holds in (11), then $v(\theta-\sigma(\beta))=v(\theta-\sigma(\alpha))$, for all $\sigma\in G$. In particular, for $\sigma=1$ we deduce $v(\theta-\beta)=v(\theta-\alpha)$, which implies

[TABLE]

because $\alpha,\,\theta$ is a distinguished pair. This proves (ii). ∎

Lemma 2.4**.**

Let $\phi,\,F\in K[x]$ be monic, irreducible polynomials with $\deg(\phi)<\deg(F)$. Then, for $\phi,\,F$ to be a distinguished pair it suffices to check that the two conditions:

1. (i) 

$0<\deg(g)<\deg(F)\ \,\Longrightarrow\,\ v(g(\theta))/\deg(g)\,\leq\,v(\phi(\theta))/\deg(\phi)$, 2. (ii)

$v(g(\theta))/\deg(g)\,=\,v(\phi(\theta))/\deg(\phi)\ \,\Longrightarrow\,\ \deg(g)\geq\deg(F)$,

hold for all monic, irreducible and separable polynomials $g\in K[x]$.

Proof.

Let us first show that if conditions (i), (ii) hold for all monic irreducible polynomials in $K[x]$, then both conditions hold for all monic polynomials.

Let $g=h_{1}\cdots h_{t}$ be a product of monic (not necessarily different) irreducible polynomials. Clearly, the average of the values $v(\theta-\beta)$ for $\beta\in\operatorname{Z}(g)$ is less than, or equal to, the maximum of the averages of the values $v(\theta-\beta)$, taken on the subsets $\operatorname{Z}(g)=\operatorname{Z}(h_{1})\cup\cdots\cup\operatorname{Z}(h_{t}).$ In other words,

[TABLE]

Therefore, (i) and (ii) hold for $g$ if they hold for $h_{1},\dots,h_{t}$.

Finally, let us show that if conditions (i), (ii) hold for all monic, irreducible separable polynomials, then both conditions hold for all monic, irreducible polynomials.

Let $g\in K[x]$ be monic and irreducible, but inseparable. Let $g_{\mbox{\tiny sep}}=g+\pi x$, for $\pi\in K^{*}$ with $v(\pi)$ sufficiently large. As mentioned in the proof of Lemma 2.1, $g_{\mbox{\tiny sep}}$ is a separable polynomial of the same degree. Since (i) and (ii) hold for all irreducible factors of $g_{\mbox{\tiny sep}}$, they hold for $g_{\mbox{\tiny sep}}$ too. Hence, if $v(\pi)$ is sufficiently large, both conditions hold for $g$. ∎

2.2. Complete distinguished chains

Definition 2.5**.**

Let $\alpha_{0},\alpha_{1},\dots,\alpha_{r},\theta=\alpha_{r+1}\in\overline{K}$ be algebraic elements such that

[TABLE]

We say that $[\alpha_{0},\alpha_{1},\dots,\alpha_{r}]$ is a complete distinguished chain for $\theta$ if $\alpha_{i},\,\alpha_{i+1}$ is a distinguished pair, for all $0\leq i\leq r$.

The next result follows immediately from Theorem 2.3.

Theorem 2.6**.**

Let $F\in K[x]$ be the minimal polynomial of $\theta\in\overline{K}\setminus K$ over $K$.

1. (1)

Let $[\phi_{0},\dots,\phi_{r}]$ be an Okutsu frame of $F$. Take $\alpha_{i}\in\operatorname{Z}(\phi_{i})$ such that

[TABLE]

Then, $[\alpha_{0},\dots,\alpha_{r}]$ is a complete distinguished chain for $\theta$. 2. (2)

Let $[\alpha_{0},\dots,\alpha_{r}]$ be a complete distinguished chain for $\theta$. Let $\phi_{0},\dots,\phi_{r}$ be the minimal polynomials of $\alpha_{0},\dots,\alpha_{r}$ over $K$, respectively.

Then, $[\phi_{0},\dots,\phi_{r}]$ is an Okutsu frame of $f$.

The next result follows immediately from Theorems 1.6 and 2.6.

Theorem 2.7** (Aghigh-Khanduja [1, 2]).**

An algebraic element $\theta\in\overline{K}$ admits a complete distinguished chain over $K$ if and only if it is defectless over $K$.

Definition 2.8**.**

Let $\alpha_{0},\alpha_{1},\dots,\alpha_{r},\theta=\alpha_{r+1}\in\overline{K}$ be algebraic elements such that

[TABLE]

We say that $[\alpha_{0},\alpha_{1},\dots,\alpha_{r}]$ is a complete Okutsu sequence for $\theta$ if the following conditions hold for all $\beta\in\overline{K}$ and all $0\leq i\leq r$:

1. (1)

$\deg_{K}(\beta)<\deg_{K}(\alpha_{i+1})\ \,\Longrightarrow\,\ v(\theta-\beta)\leq v(\theta-\alpha_{i})$. 2. (2)

$\deg_{K}(\beta)<\deg_{K}(\alpha_{i})\ \,\Longrightarrow\,\ v(\theta-\beta)<v(\theta-\alpha_{i})$.

Given a distinguished pair $\alpha,\,\theta$, we have

[TABLE]

Indeed, by the definition of distinguished pair, $v(\theta-\beta)<v(\theta-\alpha)$. This remark will be useful to compare Okutsu sequences and complete distinguished chains.

Lemma 2.9**.**

A sequence $[\alpha_{0},\alpha_{1},\dots,\alpha_{r}]$ of elements in $\overline{K}$ is a complete distinguished chain for $\theta=\alpha_{r+1}$ if and only if it is a complete Okutsu sequence for $\theta$.

Proof.

Let $\beta\in\overline{K}$ with $\deg_{K}(\beta)<\deg_{K}(\theta)$. Suppose that $[\alpha_{0},\alpha_{1},\dots,\alpha_{r}]$ is a complete distinguished chain for $\theta$. By definition, for all $0\leq i\leq r$, it holds:

1. (i) 

$\deg_{K}(\beta)<\deg_{K}(\alpha_{i+1})\ \,\Longrightarrow\,\ v(\alpha_{i+1}-\beta)\leq v(\alpha_{i+1}-\alpha_{i})$. 2. (ii)

$\deg_{K}(\beta)<\deg_{K}(\alpha_{i})\ \,\Longrightarrow\,\ v(\alpha_{i+1}-\beta)<v(\alpha_{i+1}-\alpha_{i})$.

If $i=r$, then $\alpha_{i+1}=\theta$. If $i<r$, we have that $v(\theta-\beta)=v(\alpha_{i+1}-\beta)$. Thus, in both cases, the conditions of Definition 2.8 coincide with (i) and (ii). Consequently, $[\alpha_{0},\alpha_{1},\dots,\alpha_{r}]$ is an Okutsu sequence for $\theta$.

Conversely, suppose that $[\alpha_{0},\alpha_{1},\dots,\alpha_{r}]$ is an Okutsu sequence for $\theta$. The conditions of Definition 2.8 for $i=r$ show that $\alpha_{r},\,\theta$ is a distinguished pair. By (12),

[TABLE]

for all $0\leq j<r$ and all $\beta\in\overline{K}$ with $\deg_{K}(\beta)<\deg_{K}(\alpha_{j+1})$. Therefore, the sequence $[\alpha_{0},\dots,\alpha_{r-1}]$ is a complete Okutsu sequence for $\alpha_{r}$. The previous argument shows that $\alpha_{r-1},\,\alpha_{r}$ is a distinguished pair. An iterate argument proves that $[\alpha_{0},\alpha_{1},\dots,\alpha_{r}]$ is a complete distinguished chain for $\theta$. ∎

If $[\alpha_{0},\ldots,\alpha_{r}]$ is a complete Okutsu sequence for $\theta\in\overline{K}$, then $[\alpha_{0},\ldots,\alpha_{i}]$ is a complete Okutsu sequence for $\alpha_{i+1}$, for all $1\leq i<r$, because this property is obviously true for complete distinguished chains.

3. Main invariant of quasi-tame algebraic elements

In this section, we compute several invariants attached to quasi-tame algebraic elements. To this purpose, Okutsu sequences are a more feasible tool than complete distinguished chains.

Definition 3.1**.**

Let $\theta=\alpha_{r+1}\in\overline{K}$ be defectless, and let $[\alpha_{0},\ldots,\alpha_{r}]$ be a complete Okutsu sequence for $\theta$. Denote $L=K(\theta)$ and let $k_{L}$ be the residue class field of $(L,v)$.

We say that $\theta\in\overline{K}$ is tame if

• •

$k_{L}/k$* is separable, and*

• •

the ramification index $e(L/K)$ is not divisible by $\operatorname{char}(K)$.

We say that $\theta$ is quasi-tame if it is separable and $\alpha_{r}$ is tame.

It is easy to check that a tame $\theta$ is necessarily separable over $K$.

Let $K^{s}\subset\overline{K}$ be the separable closure of $K$ in $\overline{K}$. The subgroup

[TABLE]

is the ramification subgroup of $G$. Its fixed field $K^{\mbox{\tiny ram}}=(K^{s})^{G^{\mbox{\tiny ram}}}$ is named the ramification field for the extension $K^{s}/K$. This field is the unique maximal tame extension of $K$ in $\overline{K}$. More precisely, for any algebraic extension $L/K$, the subfield $L\cap K^{\mbox{\tiny ram}}$ is the unique maximal tame extension of $K$ in $L/K$.

From now we consider $[\alpha_{0},\ldots,\alpha_{r}]$ a complete Okutsu sequence for $\theta=\alpha_{r+1}\in\overline{K}$ and we shall usually denote

[TABLE]

By Lemma 2.9, we have $\delta_{i}=v(\alpha_{i+1}-\alpha_{i})=\delta(\alpha_{i+1})$, for all $0\leq i\leq r$.

The next result is inspired in the original ideas of Okutsu [8, 5].

Proposition 3.2**.**

Let $[\alpha_{0},\ldots,\alpha_{r}]$ be a complete Okutsu sequence for a separable $\theta=\alpha_{r+1}\in K^{s}$. Consider a separable $\beta\in K^{s}$ such that

[TABLE]

for some $1\leq i\leq r+1$. Let $M/K$ be any finite Galois extension containing $K(\theta,\beta)$. Let $G=\operatorname{Gal}(M/K)$ and consider the subgroups

[TABLE]

Let $M^{H_{i}}\subset M^{\overline{H}_{i}}\subset M$ be the respective fixed fields. Finally, let $V$ be the maximal tame subextension of $K(\beta)/K$. Then,

[TABLE]

Moreover, if $v(\theta-\beta)=\delta_{i}$ then $V\subset M^{H_{i}}\subset M^{\overline{H}_{i}}\subset K(\theta)\cap K(\beta)$.

Proof.

First, let us show that $M^{H_{i}}\subset K(\theta)\cap K(\beta)$. For this, it suffices to show that all $\sigma\in G$ fixing $\theta$ or $\beta$ belong to $H_{i}$.

If $\sigma(\theta)=\theta$, then $\sigma\in H_{i}$ because $v(\theta-\sigma(\theta))=\infty>\delta_{i-1}$. If $\sigma(\beta)=\beta$, then $v(\sigma(\theta)-\beta)=v(\sigma(\theta)-\sigma(\beta))=v(\theta-\beta)>\delta_{i-1}.$ Thus,

[TABLE]

In the case $v(\theta-\beta)=\delta_{i}$, the same argument shows that $M^{\overline{H}_{i}}\subset K(\theta)\cap K(\beta)$.

Finally let us prove that $V\subset M^{H_{i}}$. Since $V$ is the maximal tame extension of $K(\beta)$, we have that $V=K^{\mbox{\tiny ram}}\cap K(\beta)$, so we must prove that

[TABLE]

Take $\sigma\in H_{i}$. Any $c\in K(\beta)^{*}$ can be written as $c=g(\beta)$ for some $g\in K[x]$ with $\deg(g)<m_{i}$. By the minimality of $m_{i}$, for any root $\xi$ of $g$ we have $v(\theta-\xi)\leq\delta_{i-1}$. Hence, $v(\beta-\xi)=\operatorname{Min}\left\{v(\beta-\theta),\,v(\theta-\xi)\right\}=v(\theta-\xi)\leq\delta_{i-1}$.

Write $g(x)=a\prod_{\xi\in\operatorname{Z}(g)}(x-\xi)$. Then,

[TABLE]

Since $\sigma\in H_{i}$, we have

[TABLE]

Since $v(\beta-\xi)=v(\theta-\xi)\leq\delta_{i-1}$, this implies $v((\sigma(\beta)-\beta)/(\beta-\xi))>0$. Hence,

[TABLE]

This proves that $V\subset M^{H_{i}}$. ∎

Lemma 3.3**.**

Let $[\alpha_{0},\ldots,\alpha_{r}]$ be a complete Okutsu sequence for $\theta\in\overline{K}$.

1. (1)

If $F\in K[x]$ is the minimal polynomial of $\theta$, then $v(\theta-\theta^{\prime})\geq v(\theta-\alpha_{0})$, for all $\theta^{\prime}\in\operatorname{Z}(F)$. 2. (2)

If $\theta$ is quasi-tame over $K$, then $\alpha_{1},\ldots,\alpha_{r}$ are tame over $K$.

Proof.

Since $\alpha_{0}\in K$, we have $v(\theta-\alpha_{0})=v(\theta^{\prime}-\alpha_{0})$ for all $\theta^{\prime}\in Z(F)$, by the henselian property. This proves (1):

[TABLE]

By Lemma 2.9, $[\alpha_{0},\ldots,\alpha_{i-1}]$ is a complete Okutsu sequence for $\alpha_{i}$. Hence, all $\alpha_{i}$ are defectless by Theorem 2.7.

As indicated in (4), we have a tower of finite extensions of $k$:

[TABLE]

Thus, the assumption that $k_{\phi_{r}}/k$ is separable implies that all $k_{\phi_{i}}/k$ are separable too.

Finally, (5) shows that

[TABLE]

Thus, if $e(\phi_{r})$ is not divisible by the characteristic of $K$, all ramification indices $e(\phi_{i})$ have the same property. This proves that $\alpha_{1},\dots,\alpha_{r}$ are tame. ∎

Theorem 3.4**.**

Let $\theta\in\overline{K}$ be quasi-tame of degree $n=\deg_{K}(\theta)>1$. Let $[\alpha_{0},\ldots,\alpha_{r}]$ be a complete Okutsu sequence for $\theta=\alpha_{r+1}$, and denote

[TABLE]

Then, it holds:

1. (1)

$K=K(\alpha_{0})\subset K(\alpha_{1})\subset\cdots\subset K(\alpha_{r})\subset K(\theta)$. 2. (2)

The following multisets of cardinality $n-1$ coincide:

[TABLE]

where  $t_{i}=(n/m_{i})-(n/m_{i+1})$  for all $0\leq i\leq r$. 3. (3)

$\delta(\theta)=\omega(\theta)=\delta_{r}$.

Proof.

Let $M/K$ be a finite Galois extension of $K$ containing $K(\theta,\alpha_{1},\dots,\alpha_{r})$, and denote $G=\operatorname{Gal}(M/K)$.

Fix an index $0\leq i\leq r$. Since $\deg_{K}(\alpha_{i})=m_{i}$ and $v(\theta-\alpha_{i})=\delta_{i}$, Proposition 3.2 applied to $\beta=\alpha_{i}$ shows that $V_{i}\subset M^{H_{i}}\subset M^{\overline{H}_{i}}\subset K(\alpha_{i})\cap K(\theta),$ where $V_{i}$ is the maximal tame subextension of $K(\alpha_{i})$.

By Lemma 3.3, $K(\alpha_{i})/K$ is tame, so that $V_{i}=K(\alpha_{i})$. Therefore,

[TABLE]

Now, denote $H_{0}:=G$ and consider the chain of subgroups

[TABLE]

The corresponding chain of fixed fields is that given in item (1).

Moreover, (13) implies

[TABLE]

so that all inclusions in the chain of subgroups are strict. Hence, for any $\sigma\in G\setminus H_{r+1}$, there exists a unique $0\leq i\leq r$ such that $\sigma\in\overline{H}_{i}=H_{i}$, and $\sigma\not\in H_{i+1}.$ If $i>0$, then $v\left(\theta-\sigma(\theta)\right)=\delta_{i}$, by the definition of the subgroups $\overline{H}_{i}$ and $H_{i+1}$.

If $i=0$, then $\sigma\not\in H_{1}$ implies $v\left(\theta-\sigma(\theta)\right)\leq\delta_{0}$. By Lemma 3.3, we deduce that $v\left(\theta-\sigma(\theta)\right)=\delta_{0}$ in this case too.

Therefore, the underlying set of the multiset $\Omega(\theta)$ is the set $\left\{\delta_{0},\dots,\delta_{r}\right\}$.

Now, it remains to find a concrete formula for the multiplicity $t_{i}$ of each value $\delta_{i}$.

Let $F\in K[x]$ be the minimal polynomial of $\theta$ over $K$. The natural action of $G$ on $\operatorname{Z}(F)$ induces a bijection:

[TABLE]

For any $0\leq i\leq r$, the restriction of this bijection to the subgroup $H_{i}/\operatorname{Gal}(M/K(\theta))$ determines a bijection:

[TABLE]

Hence, the multiplicity $t_{i}$ is equal to:

[TABLE]

This ends the proof of items (2) and (3). ∎

We end this section with an explicit formula for the main invariant $\delta(\theta)=\omega(\theta)$ in terms of the discrete invariants, described in section 1.2, attached to the inductive valuation corresponding to the minimal polynomial of $\theta$ over $K$.

Proposition 3.5**.**

With the above notation, if $\alpha_{r}$ is tame over $K$, then

[TABLE]

Proof.

Let us prove the formula by a recurrent argument on $i$. For $i=0$, we have $\phi_{0}=(x-\alpha_{0})$ and

[TABLE]

Now, suppose that $i>0$ and $\delta_{j}=\lambda_{0}+\cdots+\lambda_{j}$ for all $j<i$. Let us prove that (14) holds for $i$.

We claim that

[TABLE]

In fact, since $[\alpha_{0},\dots,\alpha_{i-1}]$ is a complete Okutsu sequence for $\alpha_{i}$, Theorem 3.4 yields an equality of multisets:

[TABLE]

for the multiplicities $t_{0},\dots,t_{i-1}$ indicated in (15).

Now, for each $\xi\in\operatorname{Z}(\phi_{i})$, $\xi\neq\alpha_{i}$, we have

[TABLE]

because $v(\theta-\alpha_{i})=\delta_{i}$, while $v(\alpha_{i}-\xi)\leq\omega(\alpha_{i})=\delta(\alpha_{i})=\delta_{i-1}<\delta_{i}.$

The equalities (16) and (17) prove the claimed identity (15), because

[TABLE]

Finally, from (15) and (6) we deduce

[TABLE]

from which we may express $\delta_{i}$ as

[TABLE]

By applying the induction hypothesis, we may express $\delta_{i}$ as a linear combination

[TABLE]

where, for $j<i$, each coefficient $a_{j}$ takes the value:

[TABLE]

This ends the proof of the proposition. ∎

The next result follows from Theorem 3.4, Proposition 3.5 and equation (6).

Corollary 3.6**.**

If $\theta\in\overline{K}$ is quasi-tame, then

[TABLE]

The last equality yields an expression for $\omega(\theta)$ which is equivalent to a formula of Brown-Merzel in [3].