## 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

### Citations:

Zbl 0535.00008; Zbl 0552.12003