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Unramified quadratic extensions of real quadratic fields, normal integral bases, and 2-adic \(L\)-functions. (English) Zbl 0891.11052

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 \(s = 1\).

MSC:

11R11 Quadratic extensions
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers

Citations:

Zbl 0081.26602
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Full Text: DOI

References:

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