A classical result on maximal valuation domains revisited (Q2744629)

A classical result of Kaplansky states that a valuation domain is linearly compact if and only if it is maximal. The main point of Kaplansky's proof is that a non linearly compact valuation domain \(R\) admits a proper immediate extension \(S.\) NEWLINENEWLINENEWLINEIn the paper under review, the author proves a very interesting result that \(R\) and \(S\) have the same residue field. The proof is well written, lucid and self-contained. The author uses the same valuation domain \(S\) as constructed by \textit{I. Kaplansky} [Duke Math. J. 9, 303-321 (1942; Zbl 0063.03135)].