Tame Valued Fields

• Tame valued fields are henselian fields where every finite extension is defectless, with ramification indices coprime to the residue characteristic.
• They exhibit a robust Galois structure by ensuring trivial wild inertia, leading to controlled tame ramification in all algebraic extensions.
• Their model theory satisfies strong AKE principles, contributing to advances in quantifier elimination and decidability in valued field settings.

A tame valued field is a central object in valuation theory and model theory, refining the structure of henselian fields by tightly constraining ramification and defect phenomena. A valued field $(K,v)$ is called tame if it is henselian and, in every finite algebraic extension, ramification indices are prime to the residue characteristic, residue extensions are separable, and all such extensions are defectless. This notion connects sources in classical ramification theory, model theory, and singularity theory, and serves as a foundational class in the study of nontrivially valued algebraic and non-archimedean fields (Kuhlmann, 6 Dec 2025, Kuhlmann, 2013, Rzepka et al., 2022).

1. Definitions and Fundamental Characterizations

Let $(K,v)$ be a valued field with valuation ring $O_K$, value group $vK$, and residue field $Kv$. The field is henselian if the valuation uniquely extends to all algebraic extensions. For a finite extension $(L|K,v)$, write

$d(L|K,v) = \frac{[L:K]}{(vL:vK)\,[Lv:Kv]}$

for the defect, where $p = \mathrm{char}(Kv)$, and $e=(vL:vK)$, $f=[Lv:Kv]$. The field $(K,v)$ is defectless if all its finite extensions satisfy $d=1$.

A finite extension is called tame if:

• (T1) The ramification index $(vL:vK)$ is coprime to $p$,
• (T2) The residue extension $Lv/Kv$ is separable,
• (T3) The extension is defectless: $d(L|K,v)=1$.

A henselian $(K,v)$ is called a tame valued field if every finite extension is tame. Equivalently, in residue characteristic $p>0$, $(K,v)$ is tame if and only if:

• (TF1) $vK$ is $p$-divisible,
• (TF2) $Kv$ is perfect,
• (TF3) $(K,v)$ is algebraically maximal (admits no nontrivial immediate algebraic extensions) (Kuhlmann, 6 Dec 2025, Rzepka et al., 2022).

2. Algebraic, Galois, and Ramification Structure

A tame field admits a deep Galois-theoretic description. The algebraic closure decomposes into a maximal tame extension ($K^t$) and its purely wild complement, with Pank's theorem providing a canonical tensor splitting $K^a \cong K^t\otimes_K K_w$, where $K_w$ is any maximal purely wild extension (Temkin, 2015). Tameness ensures that the wild inertia group in the absolute Galois group is trivial, and all Galois extensions are controlled by "tame" ramification groups.

All algebraic extensions of a tame field are again tame and defectless, and the absolute ramification field, consisting of all tame Galois subextensions, coincides with the separable closure (Kuhlmann, 2013, Kuhlmann, 2017). In particular, the class of tame fields is strictly larger than Kaplansky fields, allowing residue fields with nontrivial $p$-extensions (Anscombe et al., 2014).

3. Model Theory and AKE Principles

The model theory of tame fields is governed by strong versions of the Ax–Kochen–Ershov (AKE) principles. In the two-sorted language of valued fields, the elementary theory of a tame field is concluded by those of its value group and residue field (relative model-completeness) (Kuhlmann, 6 Dec 2025, Kuhlmann, 2013). In equal characteristic (and certain mixed characteristic cases with rank 1), relative completeness and decidability in the language of valued fields is a consequence of the decidability of the side theories (Lisinski, 2021):

Principle	Reference	Conditions	Consequence
AKE completeness	(Kuhlmann, 2013, Kuhlmann, 6 Dec 2025)	Tame, equal char	Th(K,v) determined by Th(Kv), Th(vK)
Decidability of Hahn fields	(Lisinski, 2021)	Kv, vK decidable	Th($F_q((t^\Gamma))$) decidable
Model completeness	(Kuhlmann, 6 Dec 2025)	All tame fields	Model-complete relative to side theories

Open problems involve full quantifier elimination for tame (and separably tame) fields and extensions to robust expansions such as languages with distinguished constants (e.g., $\mathcal{L}_t$) (Lisinski, 2021, Kuhlmann, 6 Dec 2025).

4. Defect, Ramification, and Generalizations

Defect theory is crucial: in equal characteristic or in mixed characteristic with rank 1, the classes of tame fields and fields such that all algebraic extensions are defectless coincide (Rzepka et al., 2022). For higher rank in mixed characteristic, fields whose henselizations are "roughly tame" (i.e., whose value group has a $p$-divisible core around $v(p)$) have all algebraic extensions defectless, but not every such field is tame.

Generalizations and variants include:

• Separably tame fields: Every separable algebraic extension is tame, with ramifications in model completeness and relative quantifier elimination (Kuhlmann et al., 2014, Anscombe, 12 May 2025).
• Roughly tame fields: Only the "p-part" of the value group is required to be divisible, relevant for mixed characteristic, higher rank cases (Rzepka et al., 2022, Jahnke et al., 2016).
• Extremal fields: Characterized by the property that valuation images of all polynomials attain maxima; all value-rank-1 tame fields are extremal, but extremality includes a broader class (Anscombe et al., 2014).
• Perfectoid/deeply ramified/semitame fields: Connections to the theory of perfectoid spaces (Scholze), semiperfect quotients, and the notion of deep ramification (Kuhlmann, 6 Dec 2025).

5. Key Examples and Non-Examples

Tame field structure arises in diverse settings:

Construction	Tame iff	Reference
$(\Q_p)^{\textrm{h}}$	Always (after henselization)	(Kuhlmann, 6 Dec 2025)
$k((t^\Gamma))$	k perfect, $\Gamma$ $p$-divisible	(Kuhlmann, 6 Dec 2025)
$\F_{q}((t^{1/p^\infty}))$	Always	(Souza, 1 Jul 2024)
$\F_p((t))$, $p>0$	Never (unless extended by $p$-roots)	(Souza, 1 Jul 2024)
Algebraically closed valued fields	Always	(Kuhlmann, 2013)

Non-examples: Non-$p$-divisible value group (e.g., $\F_p((t))$), or imperfect residue field, or occurrence of immediate Artin–Schreier or Kummer defect extensions (Rzepka et al., 2022, Kuhlmann, 6 Dec 2025).

6. Valuation-Theoretic, Graded, and Key Polynomial Criteria

Modern perspectives incorporate graded ring and key polynomial criteria. A field is tame if and only if the Frobenius on its associated graded ring is surjective (encoding $p$-divisibility and perfection) and every simple algebraic extension admits a finite complete sequence of Mac Lane–Vaquié key polynomials (Souza, 1 Jul 2024). This criterion unifies traditional, algebraic, and valuation-theoretic perspectives, and connects the resolution of singularities, local uniformization, and elimination of wild ramification (Kuhlmann, 2017, Kuhlmann et al., 2013).

7. Open Problems and Research Directions

Outstanding questions include:

• Achieving full quantifier elimination in the language of valued fields and suitable expansions,
• Decidability and explicit axiomatizations for mixed characteristic tame fields and fields like $\F_p((t))$,
• Extension of model-theoretic techniques to perfectoid, deeply ramified, and semitame fields,
• Precise structure of defect extensions and the behavior of extremality under compositions and coarsenings,
• Classification of fields whose all algebraic extensions have "independent" defect.

The study of tame valued fields remains fundamental in the tension between arithmetic structure (defect and ramification) and the quest for robust model-theoretic frameworks, with implications for the resolution of singularities, large fields, valued function field geometry, and the model theory of nonarchimedean structures (Kuhlmann, 6 Dec 2025, Rzepka et al., 2022, Kuhlmann, 2013, Souza, 1 Jul 2024).