Srivastav, A.; Venkataraman, S. Unramified quadratic extensions of real quadratic fields, normal integral bases, and 2-adic \(L\)-functions. (English) Zbl 0891.11052 J. Number Theory 67, No. 2, 139-145 (1997). The authors deduce from a result of H. B. Mann [On integral bases, Proc. Am. Math. Soc. 9, 167–172 (1958; Zbl 0081.26602)] that a number field \(F\) has an unramified quadratic extension with a normal integral basis if and only if \(F\) contains a non-square unit congruent to a square modulo \(4\). For real quadratic number fields \(F\), this condition is then expressed in terms of the \(2\)-adic logarithm of the fundamental unit, thus relating the existence of such extensions to the value of the \(2\)-adic \(L\)-function for \(F\) at \(s = 1\). Reviewer: F.Lemmermeyer (Saarbrücken) Cited in 3 Documents MSC: 11R11 Quadratic extensions 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers Keywords:normal integral basis; quadratic extension; fundamental unit; \(2\)-adic \(L\)-function Citations:Zbl 0081.26602 PDFBibTeX XMLCite \textit{A. Srivastav} and \textit{S. Venkataraman}, J. Number Theory 67, No. 2, 139--145 (1997; Zbl 0891.11052) Full Text: DOI References: [1] Brinkhuis, J., On the Galois module structure over CM-fields, Manuscripta Math., 75, 333-347 (1992) · Zbl 0757.11044 [2] Mann, H. B., On integral bases, Proc. Amer. Math. Soc., 9, 162-172 (1958) · Zbl 0081.26602 [3] Srivastav, A.; Venkataraman, S., Relative Galois module structure of quadratic extensions, Indian J. Pure Appl. Math., 25, 473-488 (1994) · Zbl 0804.11063 [4] Taylor, M. J., On Frölich’s conjecture for ring of integers of finite extensions, Invent. Math., 63, 41-79 (1981) · Zbl 0469.12003 [5] Taylor, M. J., The Galois module structure of certain arithmetic principal homogeneous spaces, J. Algebra, 153, 203-214 (1992) · Zbl 0776.11065 [6] Washington, L. C., Cyclotomic Fields. Cyclotomic Fields, Graduate Texts in mathematics, 83 (1982), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg 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.