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Some remarks on Leopoldt’s conjecture. (English) Zbl 0802.11042

Let \(p\) be an odd prime. For an algebraic number field \(k\) of finite degree over \(\mathbb{Q}\) let \(E_ k\) be the group of units of \(k\) and for a positive integer \(m\) let \(E_ k (p^ m)= \{\varepsilon\in E_ k\): \(\varepsilon\equiv 1\pmod {p^ m}\}\). The following statement is considered to be Leopoldt’s conjecture for \(k\) and \(p\) (denoted by \(\text{LC} (k,p)\)): “For each positive integer \(a\) there exists a positive integer \(m\) such that \(E_ k (p^ m) \subseteq E_ k^{p^ a}\)”.
The author investigates this conjecture \(\text{LC} (K,p)\) for a finite extension \(K/k\) such that \(p\) does not divide the degree of \(K/k\). He deals with the cases \(K/k\) is a Galois extension and a cyclic extension of prime power degree in Sections 1 and 2. The following main Theorem is proved in Section 3:
“Let \(q\) be an odd prime, \(K/k\) a finite abelian extension of degree a power of \(q\) such that \(K\not\ni e^{2\pi i/p}\), and \(q^ e\) \((e\geq 1)\) be the exponent of \(\text{Gal} (K/k)\). Assume \(p\) is a primitive root modulo \(q\) if \(e=1\), or modulo \(q^ 2\) if \(e\geq 2\), and no prime dividing \(p\) splits completely in \(K(e^{2\pi i/p} )/K\). Assume that \(\text{LC} (k,p)\) holds and there exists a positive integer \(m\) such that \[ \text{rank } {{E_ M(p^ m) M^ p} \over {M^ p}} < [M:k] (1-q^{- 1}) \] for all cyclic subextensions \(M\) of \(K/k\). Then, \(\text{LC} (K,p)\) holds.”
At the conclusion a generalization (Theorem 4.2) of a theorem of J. W. Sands [Can. Math. Bull. 31, 338-346 (1988; Zbl 0648.12004)] (Theorem 4.8) is mentioned.
Reviewer: L.Skula (Brno)

MSC:

11R20 Other abelian and metabelian extensions
11R27 Units and factorization

Citations:

Zbl 0648.12004
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References:

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