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Uniformizer of the false Tate curve extension of \(\mathbb{Q}_p\). (English) Zbl 1502.11031

Let \(p\ge 3\) be a prime number. For a positive integer \(n\), let \(\mu_{p^n}\) be the group of \(p^n\)-th roots of unity. For integers \(n\) and \(m\) with \(n\ge m\ge 0\), denote by \(K_{n,m}=\mathbb Q_p(\mu_{p^n}, p^{1/p^m})\) the false Tate curve extension of \(\mathbb Q_p\). In this paper, the canonical expansion of the primitive \(p^n\)-th root of unity \(\zeta_{p^n}\) in \(p\)-adic Mal’cev-Neumann field \(\mathbb L_p\) is studied. Using the transfinite Newton algorithm for a given polynomial \(P(T)\in\mathbb L_p[T]\), an explicit formula is proved for the first \(\aleph_0\) terms of canonical expansion of a \(p^n\)-th primitive root of unity in \(\mathbb L_p\) for every \(n\ge 2\). F. Viviani [J. Théor. Nombres Bordx. 16, No. 3, 779–816 (2004; Zbl 1075.11073)] gave a uniformizer of \(K_{1,m}\). In this paper, extending an idea of Lampert, a uniformizer of \(K_{2,m}\) is constructed with \(m\ge 1\).

MSC:

11B73 Bell and Stirling numbers
11D88 \(p\)-adic and power series fields
11P83 Partitions; congruences and congruential restrictions
11S20 Galois theory
11T22 Cyclotomy
12E30 Field arithmetic
12F10 Separable extensions, Galois theory
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)

Citations:

Zbl 1075.11073

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References:

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