$\alpha$-monogeneity of pure number fields: criterion and density

Technical Summary: $\alpha$-Monogeneity of Pure Number Fields

Problem Statement
The paper investigates the condition under which the natural order $\mathbb{Z}[\alpha]$ is the maximal order (the full ring of integers $\mathcal{O}_K$) for a pure number field $K = \mathbb{Q}(\alpha)$, where $\alpha$ is a root of the irreducible polynomial $X^n - m$ with $n \ge 2$ and $m \in \mathbb{Z} \setminus \{0\}$. This property is termed $\alpha$-monogeneity. The authors distinguish this from general field monogeneity (where $\mathcal{O}_K = \mathbb{Z}[\theta]$ for some $\theta$), noting that while $\alpha$-monogeneity implies field monogeneity, the converse does not necessarily hold for the specific generator $\alpha$. The primary goal is to establish a precise criterion for $\alpha$-monogeneity and to compute the natural density of parameters $m$ satisfying this condition within the family of pure fields of degree $n$.

Methodology
The authors employ a combination of algebraic number theory and analytic number theory:

1. Local Criteria via Dedekind's Index Theorem: The core algebraic tool is Dedekind's index criterion, which determines when a prime $p$ divides the index $[\mathcal{O}_K : \mathbb{Z}[\alpha]]$. The authors apply this to the specific polynomial $f(X) = X^n - m$.
2. Discriminant-Index Identity: They utilize the identity $\text{disc}(f) = [\mathcal{O}_K : \mathbb{Z}[\alpha]]^2 \text{disc}(\mathcal{O}_K)$ to localize the possible prime divisors of the index to the set of primes dividing $mn$.
3. Valuation Analysis: Through elementary lemmas involving $p$-adic valuations ($\nu_p$), the authors reduce the condition for index divisibility at primes $p|n$ to the behavior of the expression $m^{p} - m$ (specifically $\nu_p(m^p - m)$).
4. Density Calculations: The analytic component involves computing the natural density of integers $m$ satisfying the derived local conditions. This utilizes the Möbius function for square-free counting, the Chinese Remainder Theorem for combining local constraints, and inclusion-exclusion principles over the primes dividing $n$. The authors also refine these results to arithmetic progressions and translate parameter counts into discriminant-ordered field counts.

Key Contributions and Results

1. Necessary and Sufficient Criterion (Theorem 1.2 & 2.7):
   The paper provides a short, self-contained proof that $\mathbb{Z}[\alpha] = \mathcal{O}_K$ if and only if:

   • $m$ is square-free.
   • For every prime $p$ dividing $n$, $\nu_p(m^p - m) = 1$.

   This result specializes Smith's relative criterion for radical extensions to the pure family $X^n - m$. The authors explicitly identify the prime support of the index $[\mathcal{O}_K : \mathbb{Z}[\alpha]]$ as:
   $$\{p : \nu_p(m) \ge 2\} \cup \{p \mid n : \nu_p(m^p - m) \ge 2\}$$

2. Natural Density of $\alpha$-Monogenic Parameters (Theorem 1.1 & 3.6):
   The authors compute the natural density $\delta_n$ of parameters $m$ for which $\mathbb{Z}[\alpha]$ is maximal. For the set $M_n^+(T) = \{1 \le m \le T : X^n - m \text{ irreducible}, \mathbb{Z}[\alpha] = \mathcal{O}_K\}$, the asymptotic count is:
   $$M_n^+(X) = \delta_n X + O_n(X^{1/2})$$
   where the density is given by:
   $$\delta_n = \frac{6}{\pi^2} \prod_{p \mid n} \frac{p}{p+1}$$
   Relative to the set of square-free positive integers, the density is $\prod_{p \mid n} \frac{p}{p+1}$.

3. Arithmetic Progression Refinement (Corollary 3.10):
   The paper extends the density result to arithmetic progressions $m \equiv a \pmod q$. It provides explicit local factors $\lambda_p(q, a)$ that adjust the density based on the $p$-adic valuation of the modulus $q$ and the residue class $a$. This allows for the identification of progressions where local obstructions are automatically satisfied (Corollary 3.12).

4. Independence of Local Obstructions (Proposition 3.15):
   The authors demonstrate that the local obstruction sets at distinct primes dividing $n$ are statistically independent relative to the set of square-free integers. The density of parameters failing specific local conditions at a subset of primes $T \subseteq \{p : p \mid n\}$ is the product of the individual local failure densities.

5. Discriminant-Ordered Field Counts (Theorem 1.4 & 3.13):
   By leveraging the fact that $\mathbb{Z}[\alpha] = \mathcal{O}_K$ implies the field discriminant equals the polynomial discriminant, the authors convert parameter counts into counts of isomorphism classes of fields ordered by discriminant $Y$.
   $$N_n(Y) \sim \begin{cases} \delta_n X & \text{if } n \text{ is odd} \\ 2\delta_n X & \text{if } n \text{ is even} \end{cases}$$
   where $X = (Y/n^n)^{1/(n-1)}$.

6. Comparison with Field Monogeneity (Corollary 1.3):
   The paper clarifies the relationship between $\alpha$-monogeneity and general field monogeneity. For $n=2$, the conditions differ significantly (e.g., $m \not\equiv 1 \pmod 4$). For $n \ge 3$, the set of square-free parameters where the field is monogenic but $\mathbb{Z}[\alpha] \neq \mathcal{O}_K$ has relative density zero within the square-free integers. Thus, for $n \ge 3$, the density $\delta_n$ effectively represents the density of monogenic pure fields of degree $n$.

Significance and Claims
The authors claim that their work offers a "short proof" of the equivalence for the pure family using only Dedekind's index criterion, avoiding more complex machinery used in broader contexts. The primary significance lies in providing an exact density theory for the one-parameter family $X^n - m$. Unlike previous works that may have focused on specific degrees or used relative criteria for general radical extensions, this paper delivers a complete local-global criterion and a precise density formula dependent only on the radical of $n$.

The paper modestly notes that while the results are specific to the pure family, they resolve the question of $\alpha$-monogeneity completely for this class. Furthermore, by establishing that the "exceptional" cases (where the field is monogenic but not via $\alpha$) have density zero for $n \ge 3$, the authors assert that their density $\delta_n$ serves as the absolute positive-parameter density for genuine field monogeneity in the pure family for degrees $n \ge 3$. The work also contributes to the understanding of local obstruction patterns, showing their independence and providing tools to construct arithmetic progressions where monogeneity is guaranteed.