Pure Unital Local Principal Ideal Domains in Local Fields

Extract

The main purpose of this note is to give a characterization of p-pure unital subrings of the p-adic completion of the ring of integers R of an algebraic number field K localized at a maximal ideal p. This yields a characterization of the valued subfields of the p-adic field. In this context there turn up valuations of rational function fields in many indeterminates which seem to be new. The proof that the underlying function is indeed a valuation is quite easy here, however direct computations would involve a large amount of combinatorics. Our approach seems to fit well with Kronecker's, apparently forgotten, approach to ideal theory in rings of algebraic integers [3]. The concept of p-pure unital subrings arose from a study by the first author and A. Laroche of quasi-p-pure-injective (q.p.p.i.) abelian groups ([1], p. 582).