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Lower bounds for regulators. (English) Zbl 0549.12003

Number theory, Proc. Journ. Arith., Noordwijkerhout/Neth. 1983, Lect. Notes Math. 1068, 63-73 (1984).
The purpose of the present paper is to give some lower bounds depending on the discriminant for the regulators of some algebraic number fields of degrees 3 and 4. The main results are as follows:
Theorem 1. Let \(R\) denote the regulator of a totally real cubic field with discriminant \(D\); then \(R\geq 1/16 (\log^2(D/4))\).
For infinitely many fields the constant 1/16 cannot be replaced by a larger number.
Theorem 2. Let \(R\) denote the regulator of a totally real quartic field with discriminant \(D\) and having no quadratic subfield. Then \(R\geq (80 \sqrt{10})^{-1}\log^3(D/16)\).
In connection with these results, we note that J. H. Silverman [J. Number Theory 19, 437–442 (1984; Zbl 0552.12003)] has proved the following result, which generalizes the above results to an arbitrary algebraic number field. Let \(K\) be an algebraic number field of degree \(d\) with regulator \(R\) and absolute discriminant \(D\). Let \(r\) be the rank of the unit group of \(K\) and let \(\rho\) be the maximum of \(r\) as \(k\) ranges over all proper subfields of \(K\). Then \(R>c_d (\log (\gamma_dD)^{r-\rho}\), where we may take \(c_d=2^{-4d^2}\) and \(\gamma_d=d^{-d^{\log_28d}}\).
[For the entire collection see Zbl 0535.00008.]

MSC:

11R16 Cubic and quartic extensions
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants