Relative regulators of number fields. (English) Zbl 0945.11022

Let \(L\) be an algebraic number field. R. Zimmert [Invent. Math. 62, 367-380 (1980; Zbl 0456.12003)] found positive constants \(c_0,c_1\) such that the regulator \(\text{Reg}(L)\) satisfies \[ \text{Reg}(L)\geq c_oc_1^{[L:\mathbb Q]}. \] A.-M. Bergé and J. Martinet [Sémin. Théor. Nombres, Paris/Fr. 1987-88, Prog. Math. 81, 23-50 (1990; Zbl 0699.12014)] defined a relative regulator associated to an extension \(L/K\), which is essentially the ratio \(\text{Reg}(L)/\text{Reg}(K)\). They asked whether a result analogous to that of Zimmert holds for the relative regulator. The present authors derive such a result, i.e., they prove the following
Main Theorem: There exist positive absolute constants \(d_0,d_1\) such that for any extension \(L/K\) of number fields we have \[ \text{Reg}(L)/\text{Reg}(K)\geq\big(d_0d_1^{[L:K]}\big)^{[K:\mathbb Q]}. \]
Reviewer: V.Ennola (Turku)


11R27 Units and factorization
11R47 Other analytic theory
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