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On \(p\)-adic \(L\)-functions and normal bases of rings of integers. (English) Zbl 0815.11055

Let \(p\) be an odd prime number and \(K\) a number field containing a primitive \(p\)-th root of unity. Let \({\mathcal H} (K)\) be the subgroup of \(K^ \times/K^{\times p}\) consisting of classes \([\alpha]\) \((\alpha \in K^ \times)\) for which the extension \(K (\alpha^{1/p})/K\) is unramified, and let \({\mathcal N} (K)\) be the subgroup of \({\mathcal H} (K)\) consisting of classes \([\alpha]\) \((\in {\mathcal H} (K))\) for which the unramified cyclic extension \(K (\alpha^{1/p})/K\) has a relative normal integral basis. We investigate the Galois module structure of the quotient group \({\mathcal H} (K)/{\mathcal N} (K)\) when \(K\) runs over all layers \(k_ n\) \((n \geq 0)\) of the cyclotomic \(\mathbb{Z}_ p\)-extension \(k_ \infty/k\) of a certain imaginary abelian field \(k\). Our main theorem (Theorem 1) describes the Galois module structure of \({\mathcal H} (k_ n)/{\mathcal N} (k_ n)\) in terms of the power series associated to a certain \(p\)-adic \(L\)-function. As its application, we see that \({\mathcal H} (k_ n) = {\mathcal N} (k_ n)\) when \(n\) is sufficiently large.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R23 Iwasawa theory
11R42 Zeta functions and \(L\)-functions of number fields
11R18 Cyclotomic extensions
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