Diskret bewertete perfekte Körper mit unvollkommenem Restklassenkörper. (German) Zbl 0016.05103

Teichmüller, Oswald

J. Reine Angew. Math. 176, 141-152 (1936).

The paper begins with an exposition of the concepts of fields \(\mathbf K\) with a discrete valuation, the valuation ring \(\mathbf I\) of \(\mathbf K\), the ideals \((\pi^n)\) of \(\mathbf I\) where \(\pi\) is a prime element of \(\mathbf I\), and the residue-class field \(\mathbf F\) of \(\mathbf K\) modulo \((\pi)\). Let \(\mathbf F\) be imperfect (unvollkommen) of characteristic \(p\). Then \(\mathbf K\) may have characteristic \(p\) or zero. In the former case it is shown that \(\mathbf K\) is the field of all power series with a finite number of negative exponents in an arbitrary prime \(\pi\) over a subfield \(\mathbf T\) equivalent to \(\mathbf F\). In the latter case assume that \(\mathbf K\) is complete (perfekt). Then there exists a complete unramified subfield of \(\mathbf K\) with the same residue-class field as \(\mathbf K\). But conversely to every imperfect field \(\mathbf F\), there exists a complete unramified field of characteristic zero with \(\mathbf F\) as residue-class field,

Reviewer: A. A. Albert (Chicago)

MSC:

12J10

Valued fields

Keywords:

discretely valued perfect fields; imperfect residue class field