Lampert, David Algebraic p-adic expansions. (English) Zbl 0586.12021 J. Number Theory 23, 279-284 (1986). Let \(D_ p\) be the valuation ring in the complete algebraically closed field \({\mathbb{C}}_ p\) (with p a prime number). Let \({\mathbb{Q}}_ p^{unram}\) denote the field of the \(x\in {\mathbb{C}}_ p\) algebraic over \({\mathbb{Q}}_ p\) such that \({\mathbb{Q}}_ p(x)\cap D_ p\) is a nonramified ring extension over \({\mathbb{Q}}_ p\cap D_ p\) and let \({\mathbb{C}}_ p^{unram}\) be the p-adic completion of \({\mathbb{Q}}_ p^{unram}\). The author proves that the transcendence degree of \({\mathbb{C}}_ p^{unram}\) over \({\mathbb{Q}}_ p\) is \(2^{{\mathbb{N}}_ 0}\) and that the transcendence degree of \({\mathbb{C}}_ p\) over \({\mathbb{C}}_ p^{unram}\) is \(2^{{\mathbb{N}}_ 0}\) too. That answers a question asked in N. Koblitz’s book ”p-adic number theory, p-adic analysis and zeta- functions” (1977; Zbl 0364.12015). The author uses the p-adic ordinal series, i.e. the formal series \(\sum \xi_ i p^{r_ i}\) where the \(\xi_ i\) are roots of unity and the set of the \(r_ i\) is a well-ordered subordered-set of \({\mathbb{Q}}\). The set of the p-adic ordinal series such that all accumulation values of the set \(\{r_ i\}_ i\) are in \({\mathbb{Q}}\) is a field for the classical addition and convolution. The author proves that it is algebraically closed. Reviewer: A.Escassut Cited in 1 ReviewCited in 6 Documents MSC: 12J25 Non-Archimedean valued fields 26E30 Non-Archimedean analysis 12F20 Transcendental field extensions Keywords:valuation ring; complete algebraically closed field; nonramified ring extension; p-adic completion; transcendence degree; p-adic ordinal series Citations:Zbl 0364.12015 PDFBibTeX XMLCite \textit{D. Lampert}, J. Number Theory 23, 279--284 (1986; Zbl 0586.12021) Full Text: DOI References: [1] Koblitz, N., \((P\)-adic Number Theory, \(P\)-adic Analysis, and Zeta Functions, Vol. 58 (1977), Springer-Verlag: Springer-Verlag New York), G.T.M. · Zbl 0364.12015 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.