Algebraic p-adic expansions
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 \textit{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.
scientific article; zbMATH DE number 3939463