Minimal Mahler measures for generators of some fields (Q6044142)

For any number field \(K\), the Mahler measure \(M(K)\) can be defined as the minimum of the Mahler measures of primitive elements of \(K\). Twenty five years back, \textit{W. M. Ruppert} [Manuscr. Math. 96, No. 1, 17--22 (1998; Zbl 0899.11063)] showed that \(M(K) >> |\Delta_K|^{1/(2d-2)}\) where \(d = [K: \mathbb{Q}]\) and \(\Delta_K\) is the discriminant of \(K\). In that paper, he also asked, if, for each \(d \geq 2\), there is a constant \(k(d)\) so that for every number field \(K\) of degree \(d\), one has \(M(K) \leq k(d) |\Delta_K|^{1/(2d-2)}\). Ruppert himself proved this for \(d=2\) and later, a number of people addressed this question. \textit{J. D. Vaaler} and \textit{M. Widmer} [Math. Proc. Camb. Philos. Soc. 159, No. 3, 379--385 (2015; Zbl 1371.11141)] showed that the answer is negative for each composite \(d\) by proving that there exists a constant \(\gamma(d) > \frac{1}{2d-2}\) such that for each \(\varepsilon > 0\), there exist infinitely many number fields \(K\) of degree \(d\) satisfying \(M(K) > |\Delta_K|^{\gamma(d)- \varepsilon}\). The case \(d=5\) is also known to have a negative answer that can be deduced from Vaaler-Widmer's results and \textit{M. Bhargava}'s work [Ann. Math. (2) 172, No. 3, 1559--1591 (2010; Zbl 1220.11139)] on discriminants of qunitics. The paper under review proves that Ruppert's question has a negative answer for every odd \(d \geq 3\). More precisely, the author shows: Theorem. Let \(d \geq 3\) be odd. Then, there exist infinitely many number fields \(K\) of degree \(d\) satisfying \(M(K) > d^{-d} |\Delta_K|^{\frac{d+1}{d(2d-2)}}\). The author remarks that for composite \(d\), Vaaler and Widmer's bound is better than the one in the above theorem. One can define the Mahler measure of \(M(O_K)\) also similarly; it is the minimum of Mahler measures of algebraic integers that are primitive elements of \(K\). Clearly, \(M(K) \leq M(O_K)\). Generalizing some very recent results of \textit{L. Eldredge} and \textit{K. Petersen} [Int. J. Number Theory 18, No. 10, 2157--2169 (2022; Zbl 1521.11046)] on \(M(O_K)\) for \(d=3\), the author proves here: Theorem. For each integer \(d \geq 2\) and each \(\varepsilon > 0\), there exist infinitely many number fields \(K\) of degree \(d\) satisfying \((1- \varepsilon) |\Delta_K|^{1/d} < M(O_K) < |\Delta_K|^{1/d}\). A key auxiliary result observed and used in the proof asserts that, for any integer \(d \geq 2\) and prime \(p\), and any algebraic generator \(\alpha\) of the field \(K(p^{1/d})\), either \(\alpha\) or \(1/\alpha\) can be written as a \(\mathbb{Q}\)-linear form in \(1, p^{1/d}, \cdots, p^{m/d}\) for some \(m \geq [d/2]\) where the coefficient of \(p^{m/d}\) is non-zero. Thus, the Mahler measure \(M(\alpha) = M(1/\alpha)\) gives the exponent of \(|\Delta_K|\) in Theorem 1. Yet another auxiliary result of independent interest asserts that fr each integer \(d \geq 2\), there are infinitely many prime numbers \(p\) for which the field \(K = \mathbb{Q}(p^{1/d})\) is monogenic, with \(O_K = \mathbb{Z}[p^{1/d}]\) and \(|\Delta_K| =d^d p^{d-1}\).