Geodesic
Pure Unital Local Principal Ideal Domains in Local Fields
Canadian journal of mathematics, Tome 32 (1980) no. 2, pp. 350-353

Voir la notice de l'article provenant de la source Cambridge University Press

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). 
Benabdallah, K.; Roggenkamp, K. W. Pure Unital Local Principal Ideal Domains in Local Fields. Canadian journal of mathematics, Tome 32 (1980) no. 2, pp. 350-353. doi: 10.4153/CJM-1980-027-8
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[1] 1. Benabdallah, K. and Laroche, A., Quasi-p-pure infective groups, Can. J. Math. 29 (1977), 578–586. Google Scholar

[2] 2. Endler, O., Valuation theory (Springer-Verlag, New-York, 1972). Google Scholar

[3] 3. Kronecker, L., Vorlesungen Uber Zahlentheorie, Leipzig, 1901, BD 2, 143–241. Google Scholar

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