---

Claim. Sergio Ricardo Zapata Ceballos and Fatemeh Jalalvand, On the Squarefree Values of Degree-$2q$ Polynomials, arXiv:2608.10335v1 (11 August 2026, math.NT), Theorem 1.1: let $q$ be prime and let $L=\mathbb{Q}(\alpha)$ be a number field of degree $2q$, where $\alpha$ has monic irreducible polynomial $h\in\mathbb{Z}[x]$. If $L$ contains a Galois subextension of degree $q$ over $\mathbb{Q}$ and $\gcd\{h(n):n\in\mathbb{Z}\}$ is squarefree, then $h(n)$ is squarefree for a set of integers $n$ of positive density. The polynomial $x^4+2$ meets both hypotheses: with $q=2$, its root field contains $\mathbb{Q}(\alpha^2)=\mathbb{Q}(\sqrt{-2})$, and its values $2$ and $3$ at $0$ and $1$ make the fixed divisor $1$. So $n^4+2$ is squarefree for a positive-density set of $n$.

Covers. The third question of Problem 978, claimed yes. It also reaches the instances of the second question that its hypotheses cover: monic irreducible polynomials of degree $2q$ with a Galois subfield of degree $q$ and a squarefree fixed divisor, whose squarefree values are in particular $(2q-2)$-power-free.

Read depth. The statement of Theorem 1.1 is read in the arXiv version; the proof is not checked here.

Standing. The preprint has no journal reference and no outside review is recorded, so the claim stays claimed. The release manuscript carded as OpenAI 2026 cites the preprint and does not use it as an input; the third question is settled by OpenAI's density theorem.

Depends on. No page of this wiki.