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Kevin J. McGown

Publications and source records attributed to Kevin J. McGown.

At least 19 recordsLinked to original sources
arXiv2026

The determination of norm-Euclidean cyclic cubic fields

It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estimates which hold for all sufficiently large conductors and computational techniques. In this paper, we establish new results concerning explicit bounds for cubic non-residues and refine previous computational techniques, enabling us to completely characterise all norm-Euclidean cyclic cubic fields.

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arXiv2026

Counting metacyclic fields

By a metacyclic field $K$ we mean the Galois closure of a pure field $\mathbb{Q}(\sqrt[\ell]{D})$, $D\in\mathbb{Z}$, of odd prime degree $\ell$. Let $\mathcal{M}_\ell$ denote the collection of isomorphism classes of metacyclic fields of degree $\ell(\ell-1)$. Write $N_\ell(X)=\#\{K\in\mathcal{M}_\ell: |Δ_K|\leq X\}$ for the associated counting function, where $Δ_K$ denotes the discriminant of $K$. We show $$N_\ell(X)\sim A_\ell X^{\frac{1}{(\ell-1)^2}}(\log X)^{\ell-2}\,,$$ for an explicit constant $A_\ell$. We express $A_\ell$ as a rational number times a product over primes of a degree $\ell$ polynomial in $1/p$.

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arXiv2026

Counting number fields of fixed degree by their smallest defining polynomial

When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of $\mathrm{GL}_2 \times \mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two polynomials $f$ and $g$ in the same orbit, the roots of $f$ may be expressed as rational linear transformations of the roots of $g$; thus, they generate the same field. In this article, we show that almost all polynomials of degree $n$ with size at most $X$ can only define the same number field as another polynomial of degree $n$ with size at most $X$ if they lie in the same orbit for this group action. (Here we measure the size of polynomials by the greatest absolute value of their coefficients.) This improves on work of Bhargava, Shankar, and Wang, who proved a similar statement for a positive proportion of polynomials. Using this result, we prove that the number of degree $n$ fields such that the smallest polynomial defining the field has size at most $X$ is asymptotic to a constant times $X^{n+1}$ as long as $n\geq 3$. For $n = 2$, we obtain a precise asymptotic of the form $\frac{27}{π^2} X^2$.

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arXiv2025

Counting $C_2 \wr S_4$ fields with a power saving error term

Let $N_d(G,X)$ denote the number of degree $d$ extensions of $\mathbb{Q}$ with Galois closure $G$ and $|Δ_K|\leq X$. Malle's conjecture predicts an asymptotic of the form $N_d(G,X)\sim CX^α(\log X)^β$. Previously, Klüners proved Malle's conjecture for $G=C_2 \wr S_4$. His proof gives a power savings of $O(X^{7/8})$. We improve Klüners' result by establishing a stronger power saving error term for the count of such fields. Specifically, we show $N_8(C_2\wr S_4,X)=CX+O(X^{3/4-1/30})$. Additionally, we obtain new bounds on $N_8(G,X)$ for the groups $S_4$, $C_2^3 \rtimes S_4$, $GL_2 (\mathbb{F}_3)$, and $Q_8\rtimes S_4$ as permutation subgroups of $S_8$.

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arXiv2025

Polynomial densities and Heilbronn's criterion

Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn's criterion applies to a polynomial $f$ if it applies to the number field $K=\mathbb{Q}[x]/(f)$ generated by $f$. Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with $\gcd(p-1,n)=1$. Let ${F}_{p,n}$ denote the collection of monic polynomials $f\in\mathbb{Z}[x]$ of degree $n$ that are Eisenstein at the prime $p$. We order our polynomials by the natural height $\mathrm{Ht}(f)$. Define $δ_{p,n}(X)$ to be the proportion of polynomials $f\in {F}_{p,n}$ with $\mathrm{Ht}(f)\leq X$ for which Heilbronn's criterion applies. One has $$\liminf_{X\to\infty}δ_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\varepsilon(p)\right\}\,,$$ where $\varepsilon(p)\to 0$ and is effectively computable. In particular, the lower density tends to $1$ as $p\to\infty$ uniformly in $n$. We also give a version of this result where we weaken the condition on $\gcd(p-1,n)$. As a corollary, we show that given an integer $n\geq 2$, a positive proportion of Eisenstein polynomials of degree $n$ fail to generate norm-Euclidean fields.

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arXiv2025

The average genus number for pure fields of prime degree

Let $\ell\geq 5$ be prime. Let $\mathcal{F}_\ell$ be the collection of (isomorphism classes of) pure number fields $\mathbb{Q}(\sqrt[\ell]{a})$ of degree $\ell$, ordered by the absolute value of their discriminant. In 2018, Benli proved a counting theorem for $\mathcal{F}_\ell$, generalizing a previous theorem of Cohen and Morra when $\ell=3$. We prove that the proportion of pure fields of degree $\ell$ with genus number one is asymptotic to $(A_\ell \log X)^{-1}$ and that the average genus number for pure fields of degree $\ell$ is asymptotic to $B_\ell(\log X)^{\ell-1}$. Both $A_\ell$ and $B_\ell$ are expressed explicitly as a product over primes.

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arXiv2024

An improved error term for counting $D_4$-quartic fields

We prove that the number of quartic fields $K$ with discriminant $|Δ_K|\leq X$ whose Galois closure is $D_4$ equals $CX+O(X^{5/8+\varepsilon})$, improving the error term in a well-known result of Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting quartic dihedral extensions over an arbitrary base field.

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arXiv2023

The Counting function for Elkies primes

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_q$ where $q$ is a prime power. The Schoof--Elkies--Atkin (SEA) algorithm is a standard method for counting the number of $\mathbb{F}_q$-points on $E$. The asymptotic complexity of the SEA algorithm depends on the distribution of the so-called Elkies primes. Assuming GRH, we prove that the least Elkies prime is bounded by $(2\log 4q+4)^2$ when $q\geq 10^9$. This is the first such explicit bound in the literature. Previously, Satoh and Galbraith established an upper bound of $O((\log q)^{2+\varepsilon})$. Let $N_E(X)$ denote the number of Elkies primes less than $X$. Assuming GRH, we also show $$ N_E(X)=\frac{π(X)}{2}+O\left(\frac{\sqrt{X}(\log qX)^2}{\log X}\right)\,. $$

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arXiv2023

Counting quintic fields with genus number one

We prove several results concerning genus numbers of quintic fields: we compute the proportion of quintic fields with genus number one; we prove that a positive proportion of quintic fields have arbitrarily large genus number; and we compute the average genus number of quintic fields. All of these results also hold when restricted to $S_5$-quintic fields only.

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arXiv2023

Computation of the least primitive root

Let $g(p)$ denote the least primitive root modulo $p$, and $h(p)$ the least primitive root modulo $p^2$. We computed $g(p)$ and $h(p)$ for all primes $p\le 10^{16}$. Here we present the results of that computation and prove three theorems as a consequence. In particular, we show that $g(p) 3$ and that $h(p)<p^{2/3}$ for all primes $p$.

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arXiv2021

On abelian $\ell$-towers of multigraphs III

Let $\ell$ be a rational prime. Previously, abelian $\ell$-towers of multigraphs were introduced which are analogous to $\Z_{\ell}$-extensions of number fields. It was shown that for towers of bouquets, the growth of the $\ell$-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for $\Z_{\ell}$-extensions of number fields). In this paper, we extend this result to abelian $\ell$-towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued polynomials to construct power series with coefficients in $\Z_\ell$ arising from cyclotomic number fields, different than the power series appearing in the prequel. This allows us to study the special value at $u=1$ of the Artin--Ihara $L$-function, when the base multigraph is not necessarily a bouquet.

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arXiv2021

On abelian $\ell$-towers of multigraphs II

Let $\ell$ be a rational prime. Previously, abelian $\ell$-towers of multigraphs were introduced which are analogous to $\mathbb{Z}_{\ell}$-extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the $\ell$-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for $\mathbb{Z}_{\ell}$-extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian $\ell$-towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in $\mathbb{Z}_{\ell}$ and then study the special value at $s=1$ of the Artin-Ihara $L$-function $\ell$-adically.

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arXiv2019

Explicit bounds for small prime nonresidues

Let $χ$ be a Dirichlet character modulo a prime~$p$. We give explicit upper bounds on $q_1 0$ such that $q_n\leq Cp^{\frac{1}{4}}(\log p)^{\frac{n+1}{2}}$ whenever $n\leq n_0$ and $p\geq p_0$.

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arXiv2019

Explicit upper bounds on the least primitive root

We give a method for producing explicit bounds on $g(p)$, the least primitive root modulo $p$. Using our method we show that $g(p)<2r\,2^{rω(p-1)}\,p^{\frac{1}{4}+\frac{1}{4r}}$ for $p>10^{56}$ where $r\geq 2$ is an integer parameter. This result beats existing bounds that rely on explicit versions of the Burgess inequality. Our main result allows one to derive bounds of differing shapes for various ranges of $p$. For example, our method also allows us to show that $g(p)<p^{5/8}$ for all $p\geq 10^{22}$ and $g(p)<p^{1/2}$ for $p\geq 10^{56}$.

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arXiv2016

Statistics of genus numbers of cubic fields

We prove that approximately $96.23\%$ of cubic fields, ordered by discriminant, have genus number one, and we compute the exact proportion of cubic fields with a given genus number. We also compute the average genus number. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean.

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arXiv2016

The Euclidean algorithm in quintic and septic cyclic fields

Conditionally on the Generalized Riemann Hypothesis (GRH), we prove the following results: (1) a cyclic number field of degree $5$ is norm-Euclidean if and only if $Δ=11^4,31^4,41^4$; (2) a cyclic number field of degree $7$ is norm-Euclidean if and only if $Δ=29^6,43^6$; (3) there are no norm-Euclidean cyclic number fields of degrees $19$, $31$, $37$, $43$, $47$, $59$, $67$, $71$, $73$, $79$, $97$. Our proofs contain a large computational component, including the calculation of the Euclidean minimum in some cases; the correctness of these calculations does not depend upon the GRH. Finally, we improve on what is known unconditionally in the cubic case by showing that any norm-Euclidean cyclic cubic field must have conductor $f\leq 157$ except possibly when $f\in(2\cdot 10^{14}, 10^{50})$.

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arXiv2013

On the S-Euclidean minimum of an ideal class

We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. We also give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms. The proof is self-contained but uses ideas from ergodic theory and topological dynamics, particularly those of Berend.

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