Irreducibility in approximation couples
For a pair \(A\subset \overline{A}\) of commutative rings such that \(A\) has a linear Hausdorff topology given by a filtration of ideals and \(A\) is dense in \(\overline{A}\), we say that the couple \((A, \overline{A})\) has the Artin approximation property if for any solution in \(\overline{A}\) of a system of polynomials over \(A\) one can find an arbitrarily close approximation in \(A\) which is again a solution. This paper will prove the following Theorem: If \(A\subset \overline{A}\) are integral domains with the approximation property and \(A\) is complete and Hilbertian then \(\overline{A}\) is Hilbertian. The proof is short and elementary.
scientific article; zbMATH DE number 508814