Algebraically Maximal Discretely Valued Field

• Algebraically maximal discretely valued fields are valued fields with a discrete valuation group isomorphic to ℤ that admit no nontrivial immediate algebraic extensions.
• They play a crucial role in valuation theory by highlighting cases where henselian fields display defects in both inseparable and separable extensions.
• These fields provide practical insights into the interplay between valuation rings, residue fields, and the failure of the Fundamental Equality in unibranched extensions.

An algebraically maximal discretely valued field is a valued field $(K,v)$ with discrete valuation group $vK\cong\mathbb{Z}$ such that it admits no nontrivial immediate algebraic extensions. These fields are of fundamental importance in valuation theory, as they are, by definition, henselian and maximal with respect to immediate algebraic extension. The theory of defect and defectless fields, as well as the construction of non-defectless algebraically maximal valued fields, elucidates subtle behavior in positive characteristic and higher rank, with significant implications for the structure of field extensions and the classification of henselian fields (Kuhlmann, 8 Jan 2026).

1. Core Definitions and Characterization

Let $(K,v)$ be a valued field, where $O_K=\{x\in K\mid v(x)\ge0\}$ is the valuation ring, $\mathfrak{m}_K=\{x\in K\mid v(x)>0\}$ its maximal ideal, $vK=\{v(x)\mid x\in K^{\times}\}$ the value group, and $Kv=O_K/\mathfrak{m}_K$ the residue field. The valuation is discrete if $vK\cong\mathbb{Z}$. An algebraic extension $(L|K, v)$ is immediate if $vL=vK$ and $Lv=Kv$. The field $(K,v)$ is algebraically maximal if it admits no nontrivial immediate algebraic extension. For discretely valued fields, algebraic maximality coincides with henselianity and maximality among immediate extensions.

2. The Delon–Kuhlmann Example in Characteristic $p>0$

A paradigm for such fields is the Delon–Kuhlmann construction in characteristic $p>0$. Consider $k((t))$ with $k={\mathbf F}_p$ and the $t$-adic valuation $$v_t\bigl(\sum_i a_i t^i\bigr)=\min\{i\mid a_i\neq0\}$$. Choose algebraically independent indeterminates $x, y$ over $k(t)$ and set $s = x^p + t\,y^p$, $K = k(t, s)\subseteq k((t))$. Let $L_0$ be the relative algebraic closure of $K$ in $k((t))$.

• $(L_0, v_t)$ is discretely valued with $v_tL_0=\mathbb{Z}$ and $L_0v_t=k$.
• It is henselian and algebraically maximal.
• The extension $L_0^{1/p}=L_0(t^{1/p}, s^{1/p})$ satisfies $[L_0^{1/p}:L_0]=p^2$, $v_tL_0^{1/p}$ is an index-$p$ extension of $v_tL_0$, and $L_0^{1/p}v_t = L_0v_t$.

Applying the Lemma of Ostrowski for any unibranched extension $E|K$, $[E:K]=e(E/K)\,f(E/K)\,d(E/K),$ where $e=(vE:vK)$, $f=(Ev:Kv)$, and $d(E/K)$ is the defect. For $L_0^{1/p}|L_0$, $e=p$, $f=1$, and so $d=p$:

$d = \frac{p^2}{p\cdot 1} = p.$

Neither of the natural degree-$p$ subextensions, $L_0(s^{1/p})$ or $L_0(t^{1/p})$, is immediate, making each defectless individually, but the total extension has defect $p$. Crucially, neither subfield is algebraically maximal. This construction yields a positive-characteristic algebraically maximal discretely valued field $(L_0, v_t)$ that is not inseparably defectless.

3. Defect, Defectless Fields, and the Fundamental Equality

For a finite unibranched extension $(L|K, v)$, the ramification index

$e := (vL:vK),$

the inertia degree

$f := [Lv:Kv],$

and the defect

$d(L|K, v) = \frac{[L:K]}{e\,f} = \tilde p^{\nu},$

where $\tilde{p}=\mathrm{char}(Kv)$ if positive, and $\tilde{p}=1$ otherwise. The field $(K,v)$ is defectless if every finite unibranched extension has defect $d=1$. The failure of the Fundamental Equality, $[L:K]=ef$, is witnessed when $d>1$, signifying the presence of defect. The Delon–Kuhlmann example offers explicit evidence of algebraically maximal fields that are not defectless.

4. Separable Defect Extensions in Higher Rank

To extend the construction to separable defect extensions, rank-2 fields are constructed by introducing a second discrete valuation $w$ independent of $v_t$ with $Lw=L_0$. Forming the composed valuation $v=w\circ v_t$ on $L$ yields $v(x) = (w(x), v_t(xw))\in wL \times v_tL_0$ with lexicographic order. The resulting field $(L, v)$ is algebraically maximal of rank two, with residue $Lv=L_0v_t=k$ and value group $vL\cong wL\oplus\mathbb{Z}$.

Separable degree-$p$ subextensions of $L_0^{1/p}$ are lifted by choosing elements $a, b\in L^{\mathrm{sep}}$ satisfying $a^p = c + r a$, $b^p = d + r b$, with suitable $c, d\in L$ and $r\in L$ (where $w(r)>0$). The minimal polynomials $X^p - rX - c$ and $X^p - rX - d$ are separable, so $[L(a):L]=[L(b):L]=p$, and $L(a) v = L_0(a_0)$, $L(b) v = L_0(b_0)$. The compositum $L(a,b)$ then satisfies $[L(a,b):L]=p^2$, $vL(a,b)=vL$, $L(a,b)v=k$, so $e=p$, $f=1$, and defect $d=\frac{p^2}{p\cdot 1}=p$. Thus, degree $p^2$ separable extensions with defect $p$ are realized in both characteristic $(p,p)$ and $(0,p)$ settings for fields of rank two.

5. Implications for Valued-Field Theory

The constructions above demonstrate:

• Algebraic maximality does not imply defectlessness, even in rank one.
• Separable defect extensions can occur in higher rank, not just purely inseparable defect.
• Algebraically maximal fields may fail to be separably defectless or stable under finite separable extension.

These phenomena indicate the limits of general induction-by-henselization arguments and underscore the subtle role of defect in valued field theory. Such observations further constrain classification results and methods in the structure theory of henselian and algebraically maximal fields.

6. Current Open Problems

A central open question is whether there exist rank-one algebraically maximal fields admitting separable defect extensions. As of the latest results, including (Kuhlmann, 8 Jan 2026), no such example is known, and the existence question remains unresolved. This problem is directly motivated by the contrast between the established constructions in higher rank and the lack of parallel results in rank one.