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An inequality relating the regulator and the discriminant of a number field. (English) Zbl 0552.12003

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.

MSC:

11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11R21 Other number fields
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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
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