Silverman, Joseph H. An inequality relating the regulator and the discriminant of a number field. (English) Zbl 0552.12003 J. Number Theory 19, 437-442 (1984). Let \(K\) be a number field of degree \(d\) with regulator \(R\) and absolute discriminant \(D\). Let \(r=r(K)\) be the rank of the unit group in \(K\), and let \(\rho\) be the maximum of \(r(k)\) as \(k\) ranges over proper subfields of \(K\). It is proved that \(R > c_d (\log (\gamma_dD))^{r-\rho}\) for positive constants \(c_d\), \(\gamma_d\) depending only on \(d\). The proof is simple but interesting. The author makes the plausible conjecture that the exponent \(r-\rho\) is best possible. There are a few annoying misprints. E.g., in the definition of \(\alpha(j)\) on p. 440 inequality should be replaced by equality. Reviewer: Veikko Ennola (Turku) Cited in 6 ReviewsCited in 20 Documents MSC: 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants 11R21 Other number fields Keywords:regulator; discriminant; inequality PDFBibTeX XMLCite \textit{J. H. Silverman}, J. Number Theory 19, 437--442 (1984; Zbl 0552.12003) Full Text: DOI References: [1] T. W. Cusick; T. W. Cusick · Zbl 0549.12003 [2] Dobrowolsky, E., On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith., 34, 391-401 (1979) · Zbl 0416.12001 [3] Lang, S., (Fundamentals of Diophantine Geometry (1983), Springer: Springer New York) · Zbl 0528.14013 [4] Pohst, M., Regulatorabschätzungen für total reelle algebraische Zahlkörper, J. Number Theory, 9, 459-492 (1977) · Zbl 0366.12011 [5] Remak, R., Über Grössenbesiehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers, Compositio Math., 10, 245-285 (1952) · Zbl 0047.27202 [6] Silverman, J., The Thue equation and height functions, (Approximations-Diophantiennes et Nombres Transcendants. Approximations-Diophantiennes et Nombres Transcendants, Progress in Math. (1983), Birkhäuser: Birkhäuser Boston), 259-270 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.