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arxiv: 2606.29310 · v1 · pith:BJ66UWI5new · submitted 2026-06-28 · 🧮 math.NT

Iterated extensions and the ramification dichotomy

Mugurel Barcau , Vicen\c{t}iu Pa\c{s}ol This is my paper

Pith reviewed 2026-06-30 02:40 UTC · model grok-4.3

classification 🧮 math.NT
keywords p-adic ramificationiterated extensionsdeeply ramified extensionsperfectoid fieldsinverse branchesFrobenius polynomialsdifferent ideals
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The pith

Towers from iterated polynomial inverses over p-adic fields are either unramified or deeply ramified.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper considers finite extensions K of the p-adics together with a monic polynomial f of degree at least two whose derivative lies in the maximal ideal times the coefficient ring. A compatible sequence of preimages under f generates a tower of fields whose union is shown to satisfy a clean dichotomy: the whole tower is unramified, or else, as soon as ramification appears, the ramification indices over the maximal unramified subextensions tend to infinity while the different ideals at finite levels become unbounded. This classification directly describes the possible ramification behavior of infinite extensions built by dynamical iteration.

Core claim

Let K be a finite extension of Q_p and let f be a monic polynomial in O_K[X] of degree at least two with f' belonging to m_K O_K[X]. For any compatible inverse branch t_n satisfying f(t_{n+1})=t_n with t_0 in O_K, the fields K_n = K(t_n) and their union K_∞ satisfy that K_∞/K is either unramified or deeply ramified; once ramification occurs the ramification indices over maximal unramified subfields tend to infinity and the finite-level differents are unbounded. In the special case f ≡ X^{p^a} mod m_K the unramified alternative collapses to K_∞ = K or deep ramification. Completion of a non-unramified tower yields a perfectoid field, while examples demonstrate that the arithmetically profinite

What carries the argument

The compatible inverse branch (t_n) under f, which generates the tower K_∞ whose ramification behavior is classified by the dichotomy.

If this is right

  • Once any ramification appears, ramification indices over the maximal unramified subfields tend to infinity.
  • The different ideals of the finite extensions K_n become unbounded.
  • When f is congruent to a pure p-power modulo the maximal ideal, the tower is either equal to the base field or deeply ramified.
  • Completion of any non-unramified tower produces a perfectoid field.
  • The arithmetically profinite property need not hold for the algebraic tower even when the completion is perfectoid.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dichotomy supplies explicit algebraic constructions whose completions are perfectoid with controlled ramification growth.
  • The observed failure of the arithmetically profinite property at the algebraic level shows that completion is essential for certain profinite features to appear in these towers.

Load-bearing premise

A compatible sequence of preimages under f exists starting from an integral point, and f is monic of degree at least two with derivative divisible by the maximal ideal.

What would settle it

An explicit monic polynomial f of degree at least two satisfying the derivative condition, together with a compatible sequence t_n, such that ramification appears at some finite level but the ramification indices over the maximal unramified subextensions remain bounded for all larger n, or the different ideals of the K_n remain bounded.

read the original abstract

Let $K/\mathbb Q_p$ be finite and let $f\in\mathcal O_K[X]$ be monic, of degree at least two, with $f'(X)\in\mathfrak m_K\mathcal O_K[X]$, equivalently $\bar f\in k[X^p]$. For a compatible inverse branch $f(t_{n+1})=t_n$ with $t_0\in\mathcal O_K$, put $K_n=K(t_n)$ and $K_\infty=\bigcup_nK_n$. We prove that $K_\infty/K$ is either unramified or deeply ramified. More precisely, once ramification appears, the ramification indices over the maximal unramified subfields tend to infinity and the finite-level differents are unbounded. In the Frobenius-type case $f(X)\equiv X^{p^a}\pmod{\mathfrak m_K}$ the unramified alternative is trivial, so $K_\infty=K$ or $K_\infty/K$ is deeply ramified. After completion, the non-unramified alternative gives perfectoid fields and examples show that APF property need not hold at the algebraic level.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that for a finite extension K/Q_p and a monic f in O_K[X] of degree at least 2 with f' in m_K O_K[X] (equivalently bar f in k[X^p]), any tower K_∞ obtained from a compatible inverse branch t_n (f(t_{n+1})=t_n, t_0 in O_K) is either unramified over K or deeply ramified. In the latter case, ramification indices over maximal unramified subfields tend to infinity and finite-level differents are unbounded. The Frobenius-type case f ≡ X^{p^a} mod m_K makes the unramified alternative trivial, and completion of the non-unramified case yields perfectoid fields (though APF need not hold algebraically).

Significance. If the result holds, it supplies a clean, parameter-free dichotomy in ramification theory for iterated inverse branches under the given derivative condition, with direct implications for perfectoid fields and the distinction between algebraic and completed towers. The statement is independent of fitted quantities and offers a falsifiable classification that aligns with standard local-field tools.

minor comments (2)
  1. The abstract uses K_∞ and K_n without an explicit definition of the maximal unramified subfield in the statement of the dichotomy; a brief clarification in the introduction would aid readability.
  2. The parenthetical equivalence 'equivalently bar f in k[X^p]' is standard but could be expanded with a one-line reference to the reduction map in §1 or §2 for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our results and for the positive evaluation of their significance. The recommendation is listed as uncertain, yet the report contains no specific major comments requiring a point-by-point reply. We remain available to address any further questions the referee may have.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a ramification dichotomy for towers K_∞/K generated by compatible inverse branches of a monic polynomial f satisfying f' ∈ m_K O_K[X] (equivalently ar f ∈ k[X^p]). The proof proceeds from the standard setup of local-field ramification theory, using the given conditions on f and the branch t_n to show that either the extension remains unramified or the ramification indices and differents become unbounded. No equation or definition reduces the claimed dichotomy to a fitted quantity, self-referential construction, or load-bearing self-citation; the result is an independent theorem whose assumptions are external to the conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a theorem in algebraic number theory relying on standard background; no free parameters are introduced, no new entities are postulated, and axioms are the usual ones of p-adic ramification theory.

axioms (1)
  • standard math Standard ramification theory for finite extensions of Q_p, including definitions of ramification index, different, and maximal unramified subfield.
    Invoked throughout the statement of the dichotomy and the description of deep ramification.

pith-pipeline@v0.9.1-grok · 5735 in / 1304 out tokens · 42811 ms · 2026-06-30T02:40:28.167794+00:00 · methodology

discussion (0)

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Reference graph

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