Monogenic Fields from Polynomial Compositions with Applications

Abstract: A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}<em>K$ can be expressed as a simple ring extension $\mathbb{Z}[α]$ for some $α\in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for $[\mathbb{Z}</em>{K_i}:\mathbb{Z}[α<em>i]]=1$, where $K_i=\mathbb{Q}(α_i)$ and $α_i$ is a root of the composed polynomial $f_i(x<sup>k+b)$ for $i=1,2$. Here, $$f_1(x)=x^n+c\sum</em>{j=1}<sup>{n}(ax)<sup>{n-j}\in\mathbb{Z}[x]$$ and $$f_2(x)=x<sup>n+c\sum_{j=1}<sup>{n}a<sup>{j-1}x<sup>{n-j}\in\mathbb{Z}[x]$$ are irreducible polynomials of degree $n\ge 3$. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations.

• The paper establishes necessary and sufficient prime-by-prime index criteria for monogenity in compositions of Harrington–Jones polynomial families with binomials x^k+b, using Dedekind–Uchida methods and explicit congruence conditions.
• It derives corrected discriminant formulas for both composition families, identifies an unconditional non-monogenity obstruction when a prime divides a but not c, and constructs explicit monogenic sextics with non-squarefree discriminants.
• Assuming the abc-conjecture, it obtains a quantitative lower bound for simultaneously monogenic pairs and shows how monogenity supplies integral power-basis coordinates for solutions of related constant-coefficient differential equations.

Overview

This paper, by Jakhar, Kalwaniya, and Yadav (2605.00949), studies the monogenity of number fields defined by compositions of two explicit polynomial families with an arbitrary binomial. The base families are those introduced by Harrington and Jones:

$$f_1(x) = x^n + c\sum_{i=1}^{n}(ax)^{n-i}, \qquad f_2(x) = x^n + c\sum_{i=1}^{n}a^{i-1}x^{n-i},$$

with $n \ge 3$, and the composition is taken against $g(x) = x^k + b$. The central question is when a prime divisor $p$ of the polynomial discriminant $D_F$ fails to divide the index $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$, where $\theta$ is a root of $F = f_i \circ g$. The paper delivers necessary and sufficient conditions for both families (Theorems 1 and 2), an unconditional non-monogenity criterion, discriminant formulas for the composed polynomials, a conditional (on the $abc$-conjecture) asymptotic lower bound for pairs of simultaneously monogenic polynomials, and an application to linear differential equations whose auxiliary polynomials are these compositions.

The work generalizes the authors' earlier treatment of the same two families without composition ($k=1$, $n \ge 3$0), and the paper notes explicitly that setting $n \ge 3$1, $n \ge 3$2 recovers those prior main results with no additional hypotheses on $n \ge 3$3, $n \ge 3$4, or $n \ge 3$5.

Discriminant formulas for the compositions

A corrected version of Cullinan's formula for the discriminant of a composition is used as the technical backbone. The authors point out that Cullinan's original statement omits a sign factor $n \ge 3$6 arising in passing from $n \ge 3$7 to $n \ge 3$8; they supply the correction. Combining this with the Harrington–Jones discriminant formulas for $n \ge 3$9 and $g(x) = x^k + b$0 yields closed-form expressions:

$g(x) = x^k + b$1

and

$g(x) = x^k + b$2

These factorizations are what make the index analysis tractable: every prime dividing $g(x) = x^k + b$3 divides one of a small number of explicit integer factors, so the Dedekind–Uchida criterion can be applied prime by prime.

Index criteria

The two main theorems give, for each prime $g(x) = x^k + b$4, an if-and-only-if condition for $g(x) = x^k + b$5, organized by the divisibility pattern of $g(x) = x^k + b$6 among $g(x) = x^k + b$7, $g(x) = x^k + b$8, $g(x) = x^k + b$9, $p$0, and $p$1. The proofs proceed via the ideal-theoretic form of Dedekind's criterion: $p$2 is checked by expanding $p$3 modulo powers of $p$4 around its repeated factors.

Several structural features are worth noting. When $p$5 with $p$6, the reduction of $p$7 modulo $p$8 is a $p$9-th power of a lower-degree polynomial $D_F$0, and the index condition reduces to coprimality of $D_F$1 with $D_F$2; under additional congruences on $D_F$3 and $D_F$4 this collapses to a single non-divisibility such as $D_F$5. In the generic case $D_F$6, all nonzero repeated roots of $D_F$7 modulo $D_F$8 are forced to lie over a single residue class determined by $D_F$9, where $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$0, and the criterion becomes $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$1 together with $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$2 and $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$3. A parallel analysis for $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$4 replaces these quantities with $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$5, $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$6, and $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$7.

An immediate corollary is a clean sufficient condition: if $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$8, $[\mathbb{Z}_K : \mathbb{Z}[\theta]]$9, $\theta$0, certain squarefree conditions hold on $\theta$1, $\theta$2, and the combination $\theta$3, and a congruence condition on $\theta$4 versus $\theta$5 modulo $\theta$6 holds for primes dividing $\theta$7, then both $\theta$8 and $\theta$9 are irreducible and monogenic. Irreducibility here comes from Eisensteinity at a prime dividing $F = f_i \circ g$0.

The paper also records an unconditional obstruction: if $F = f_i \circ g$1 and $F = f_i \circ g$2, then $F = f_i \circ g$3 is non-monogenic, since $F = f_i \circ g$4 has repeated roots lying in $F = f_i \circ g$5. This constrains any family seeking simultaneous monogenicity to primes avoiding $F = f_i \circ g$6.

Counting monogenic pairs under the abc-conjecture

Assuming the $F = f_i \circ g$7-conjecture, the paper proves a lower bound for the number of parameter pairs $F = f_i \circ g$8 with $F = f_i \circ g$9, $abc$0 for which both $abc$1 and $abc$2 are monogenic. The proof combines Granville's theorem on squarefree values of polynomials (which itself relies on $abc$3) applied to $abc$4 and to the auxiliary polynomial $abc$5 defined implicitly by

$abc$6

with a Chinese remainder theorem argument enforcing the congruence conditions of the corollary above. The resulting bound is of order

$abc$7

where $abc$8 is a fixed prime, $abc$9, and $k=1$0. Two points deserve emphasis. First, the result is conditional: it depends on the $k=1$1-conjecture through Granville's theorem, and the paper does not claim an unconditional count. Second, the count concerns pairs $k=1$2 that are simultaneously monogenic, which is stronger than counting monogenic fields individually. The authors state, without proof, that analogous techniques yield a corresponding lower bound for the $k=1$3 family.

This places the paper in the line of work initiated by Kedlaya's construction of polynomials with squarefree discriminants, Jones' families of monogenic polynomials with non-squarefree discriminant, and Bhargava–Shankar–Wang's result that at least 30% of polynomials have squarefree discriminant. The contribution here is a quantitative supply of monogenic compositions — a setting where squarefreeness of $k=1$4 typically fails because of the factors $k=1$5 and $k=1$6.

Application to differential equations

The final theoretical section translates the arithmetic results into a description of solutions of constant-coefficient linear differential equations obtained by substituting $k=1$7 into the polynomial patterns of $k=1$8 and $k=1$9. If the auxiliary polynomial $n \ge 3$00 is irreducible and every prime divisor of its discriminant satisfies the relevant index criterion, then $n \ge 3$01 and each root of the auxiliary equation is an integral linear combination $n \ge 3$02. The general solution is then written as

$n \ge 3$03

with integer exponents' coefficients $n \ge 3$04 and arbitrary real constants $n \ge 3$05. The logical content is that monogenicity guarantees the roots of the characteristic equation admit coordinates in a power basis, giving an explicit integral parametrization of the exponential modes. It should be noted that the derivation is essentially formal: once irreducibility holds, the solution space of a degree-$n \ge 3$06 linear ODE is spanned by exponentials of the roots regardless of monogenicity; the arithmetic hypothesis controls the integrality of the coefficients describing those roots, not their existence.

Examples

Three concrete illustrations are given. For $n \ge 3$07, $n \ge 3$08, $n \ge 3$09, any $n \ge 3$10 with $n \ge 3$11, and $n \ge 3$12, one obtains the simultaneously monogenic pair $n \ge 3$13 and $n \ge 3$14; infinitude of admissible $n \ge 3$15 follows from Erdős' theorem on squarefree values of primitive polynomials. For the second family, $n \ge 3$16 and $n \ge 3$17 have $n \ge 3$18, and each prime is handled by the criteria, so $n \ge 3$19 is monogenic. Similarly, $n \ge 3$20 with $n \ge 3$21 is verified monogenic via case (vi) of the first main theorem, providing an explicit monogenic sextic quadrinomial with non-squarefree discriminant.

Limitations and open questions

Several caveats attach to the results. The counting theorem is conditional on the $n \ge 3$22-conjecture, and no unconditional lower bound is established for either family. The analytic section treats only the $n \ge 3$23 family in detail; the analogous statement for $n \ge 3$24 is asserted only by remark, without proof. The index criteria require irreducibility of $n \ge 3$25 as a standing hypothesis, and the paper does not characterize when $n \ge 3$26 is irreducible for general parameters beyond the Eisenstein situations used in the corollary. The differential-equation application assumes the auxiliary polynomial is irreducible and satisfies the full set of index conditions; the paper does not address reducible auxiliary equations or the effect of repeated roots of the characteristic equation on the solution basis. Finally, the criteria are stated prime by prime across six (respectively five) cases, and no unified single-condition characterization is offered.

Conclusion

The paper extends the monogenity theory of the Harrington–Jones polynomial families from the base polynomials to their compositions with arbitrary binomials $n \ge 3$27, supplying complete prime-by-prime index criteria, corrected discriminant formulas for the compositions, a non-monogenity obstruction at primes dividing $n \ge 3$28, a conditional density-type lower bound for simultaneously monogenic parameter pairs, and an integral-basis description of solutions of associated higher-order linear differential equations. The examples demonstrate that the criteria are effective in producing explicit monogenic polynomials of composite degree with non-squarefree discriminants.