Representation of Witt Vectors by Formal Power Series and its Applications
Kiyomi KANESAKA, Koji Sekiguchi · Tokyo Journal of Mathematics · 1979
K$ of characteristic $p$ by the matrix equation of the type $X^{p}=MX,$ $MeGL.(K)$ .We define in \S 1 an isomorphism $f_{u}$ of the additive group $W_{\infty}(K)$ of Witt vectors into the multiplicative 1- unit group of the formal power series ring $K[[t]]:f_{u}(\alpha+\alpha')=f_{u}(\alpha)\cdot f_{u}(\alpha^{\prime}).*)$As an application we consider in \S 2 the relation between the Witt theory and the Inaba theory.Y. Kawada and I. Satake [7] applied the residue vectors defined in Witt [10] to the class formation theory over a formal power series field $K$ in one variable with a finite constant field.In \S 3 we calculate the residue vectors by the use of the mapping $f_{u}$ defined in \S 1.Using these results, we consider in \S 4 the orthogonal pairings and the duality defined by residue vectors.In \S 5 we consider the formal power series field $K$ in one variable with a finite constant field and the cyclic extension field $L$ of order $p^{n}$ over $K$ .We calculate the ramification index and the conductor of $L$ over $K$ .