[Paper Review] Solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers

J. M. Casas, B. A. Omirov+1 more|arXiv (Cornell University)|Feb 10, 2011

advanced mathematical theories14 references11 citations

TL;DR

This paper establishes explicit solvability criteria for the equation $x^q = a$ in the field of $p$-adic numbers $\mathbb{Q}_p$ for all $q$, reducing the general case to two fundamental cases: (i) $q$ not divisible by $p$, and (ii) $q = p$. It provides constructive algorithms to find solutions when they exist and proves that every $p$-adic number admits a finite set of representations in the form $x = \varepsilon\delta y^q$, where $\varepsilon$ and $\delta$ range over specific finite sets depending on $p$. The key contribution is a complete, algorithmic characterization of $q$-th power solvability in $\mathbb{Q}_p$.

ABSTRACT

We establish the solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers, for any $q$ in two cases: (i) $q$ is not divisible by $p$; (ii) $q=p$. Using these criteria we show that any $p$-adic number can be represented in finitely many different forms and we describe the algorithms to obtain the corresponding representations. Moreover it is showed that solvability problem of $x^q=a$ for any $q$ can be reduced to the cases (i) and (ii).

Motivation & Objective

• To establish complete solvability criteria for the equation $x^q = a$ in $\mathbb{Q}_p$ for arbitrary $q$.
• To reduce the general solvability problem for $x^q = a$ to two fundamental cases: $q$ not divisible by $p$ and $q = p$.
• To provide constructive algorithms for computing solutions when they exist.
• To describe all possible representations of any $p$-adic number as $x = \varepsilon\delta y^q$, with $\varepsilon$, $\delta$ in explicitly defined finite sets.
• To extend known results on $x^2 = a$ to higher-degree equations $x^q = a$ in $p$-adic fields, relevant for classification problems in algebra.

Proposed method

• Use of $p$-adic valuation and $p$-adic expansion to analyze the structure of $x^q = a$.
• Application of discrete logarithms and primitive roots modulo $p^k$ to reduce the solvability condition to congruence conditions on indices.
• Leveraging Hensel's lemma and lifting techniques to construct solutions iteratively in the $p$-adic setting.
• Defining finite sets $\mathcal{E}_1$ and $\mathcal{E}_2$ of representatives for $p$-adic units and $p$-powers, respectively, to parametrize all $q$-th power classes.
• Reduction of the general case $q = m p^s$ with $p \nmid m$ to successive applications of the two base cases via substitution $y = x^{p^s}$, $z = x^{p^{s-1}}$, etc.
• Using Theorem 3.7 (on $p$-th power residues) and Theorem 3.2 (on $q$-th power residues for $p \nmid q$) as foundational tools for the reduction and construction.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for the equation $x^q = a$ to have a solution in $\mathbb{Q}_p$ when $q$ is not divisible by $p$?
• RQ2 What are the solvability criteria for $x^p = a$ in $\mathbb{Q}_p$, and how can solutions be explicitly constructed?
• RQ3 How can the solvability of $x^q = a$ be reduced to the two base cases $p \nmid q$ and $q = p$?
• RQ4 What is the structure of the set of all $p$-adic numbers that are $q$-th powers, and how can each $a \in \mathbb{Q}_p$ be represented as $\varepsilon\delta y^q$?
• RQ5 For a given $p$, what are the explicit finite sets $\mathcal{E}_1$ and $\mathcal{E}_2$ that parametrize the non-$q$-th power components of $p$-adic numbers?

Key findings

• For $q$ not divisible by $p$, the equation $x^q = a$ is solvable in $\mathbb{Q}_p$ if and only if the index $\mathrm{ind}_r(a)$ modulo $\varphi(p^k)$ is divisible by $\gcd(q, \varphi(p^k))$, where $r$ is a primitive root modulo $p^k$.
• For $q = p$, the equation $x^p = a$ is solvable in $\mathbb{Q}_p$ if and only if $a$ satisfies the conditions of Theorem 3.7: $\gamma(a) \geq 0$ and the leading term of $a$ modulo $p^2$ lies in a specific set of non-$p$-th power residues.
• Any $p$-adic number $a \in \mathbb{Q}_p$ can be uniquely written in the form $a = \varepsilon \delta y^p$, where $\varepsilon \in \mathcal{E}_1$ and $\delta \in \mathcal{E}_2$, with $\mathcal{E}_1$ and $\mathcal{E}_2$ being finite sets depending on $p$.
• For $p = 3$, the set $\mathcal{E}_1 = \{1, 4, 5\}$ and $\mathcal{E}_2 = \{1, 3, 9\}$, so every $a \in \mathbb{Q}_3$ has a representation $a = \varepsilon \delta y^3$ with $\varepsilon \in \mathcal{E}_1$, $\delta \in \mathcal{E}_2$.
• For $p = 5$, $\mathcal{E}_1 = \{1, 11, 12, 13, 14\}$ and $\mathcal{E}_2 = \{1, 5, 25, 125, 625\}$, so every $a \in \mathbb{Q}_5$ admits a unique representation $a = \varepsilon \delta y^5$.
• The general case $q = m p^s$ with $p \nmid m$ reduces to solving $y^m = a$ followed by $z^p = \tilde{y}$, $w^p = \tilde{z}$, etc., with solvability conditions inherited from the two base cases.