Square-free values of cubic polynomials in algebraic number fields
The author generalizes the well-known work of Hooley on square-free values of cubic polynomials to algebraic number fields. Let K be a fixed algebraic number field over the rationals of degree \(n=r_ 1+2r_ 2\) (in the usual notation) and with discriminant d. \({\mathbb{Z}}_ K\) denotes the ring of integers in K. For any real numbers \(P_ 1,...,P_ n\) with \(P_ i\geq 1\) and \(P_ k=P_{k+r_ 2}\), \(r_ i+1\leq k\leq r_ 1+r_ 2\), put \(P=P_ 1\cdot \cdot \cdot P_ n\) and let \({\mathfrak R}(P)=\{\alpha \in {\mathbb{Z}}_ K\); \(\alpha\) totally positive, \(| \alpha^{(k)}| \leq P_ k\), \(1\leq k\leq n\}\). Then we have: Theorem 1: Let \(F(x)\in {\mathbb{Z}}_ K[x]\) be an irreducible polynomial of degree \(g\geq 3\). Let \(L({\mathfrak a})\) denote the number of solutions of the congruence \(F(\alpha)\equiv 0 modulo an\) integral ideal \({\mathfrak a}\) of K. Suppose that L(\({\mathfrak p}^{g-1})<N{\mathfrak p}^{g-1}\) for all prime ideals \({\mathfrak p}\) of K. Then \[ \sum_{\alpha \in {\mathfrak R}(p),F(\alpha)(g-1)-free}= \frac{(2\pi)^{r_ 2}}{| \sqrt{d}|}P\prod_{{\mathfrak p}}(1-\frac{L({\mathfrak p}^{g-1})}{N{\mathfrak p}^{g-1}})+O(\frac{P(\log \log P)^{r_ 1+r_ 2-1}}{(\log P)^{(g- 1)/(g+1)}}), \] where the infinite product is convergent and positive. The author also proves a corresponding Theorem 2 which relates to the case when \(\alpha\) is a prime in K. In this case, g is restricted to \(g\geq 7\).
existence of infinitely many power-free algebraic integers in polynomial sequences
scientific article; zbMATH DE number 4123837