Some elementary remarks about \(n\)-local fields
Let \((K, v)\) be a valued field where \(v\) is discrete and of rank \(n\). Then \((K, v)\) is called \(n\)-local, if \((K_{i+1}, \overline {v}_i)\) is local, \(i=0, \dots, n-1\), where \(K_{i+1}= O_{i+1}/ M_{i+1}\) \(((0)= M_n \subset M_{n-1} \subset \dots \subset M_0\) are all the prime ideals of a valuation ring \(O_v\) of \(v\), \(O_{i+1}\) is the localization of \(O_v\) with respect to \(M_{i+1}\) and \(\overline {v}_i\) is the valuation on the field \(K_{i+1}\) associated with \(O_i/ M_{i+1}\)). An example of an \(n\)-local valued field is the field \(k_n= k((t_1 ))\dots ((t_n ))\) of repeated power series. If \(k\) is a perfect field of characteristic \(p>0\) the authors show a construction of a rank one and discrete valuation with residue field \(k_n\). They prove also the following theorem: If \(K\) is an \(n\)-local field such that its residue field has the property that for every natural number \(m\) there is a finite set of separable extensions of degree \(m\), then \(K\) has, for every natural number \(m\), a finite number of tamely ramified extensions of degree \(m\).
scientific article; zbMATH DE number 763289
scientific article