Properties of extensions of algebraically maximal fields
We prove analogues to the Ax-Kochen-Ershov theorem for some cases of pairs of algebraically maximal fields of residue characteristic \(p >0\). By a pair of valued fields, we mean a structure \((K\subset L,v)\), with \((L, v)\) a valued field, and \(K\) a subfield of \(L\). This pair is said to be dense if \(K\) is dense in \(L\) with the topology associated to \(v\). It is said to be separated if, for every finitely generated \(K\)-submodule \(A\) of \(L\), \(v(1-A)\) has a greatest element. Among the consequences of the results are the following two theorems: (1) Let \((K\subset L,v)\) and \((K'\subset L',v')\) be two separated pairs of algebraically maximal fields of positive characteristic (resp. unramiefied fields with perfect residue fields). Assume that the pairs of residue fields are elementarily equivalent and that the pairs of value groups are elementarily equivalent. Then \((K\subset L,v)\) and \((K'\subset L',v')\) are elementarily equivalent. (2) The result in (1) still holds for dense pairs of algebraically maximal Kaplansky fields (resp. finitely ramified fields with perfect residue fields). We generalize those results to some \(n\)-tuples \((K_1 \subset \cdots\subset K_n,v)\) of algebraically maximal fields, and to non-separated non-dense pairs. Then, we deduce from that the decidability of some theories of pairs of algebraically maximal valued fields.
pairs of valued fields of positive characteristic
scientific article; zbMATH DE number 1943363