An algebraic extension \((L|K, v)\) is called tame if the extension of \(v\) from \(K\) to \(L\) is unique and, for every finite subextension \(E|K\) of \(L|K\), the residue field extension \(Ev|Kv\) is separable, \((vE : vK)\) is not divisible by char\(Kv\) (\(vK\) the valuation group of \((K,v)\)), and \([E : K] = (vE : vK)[Ev : Kv]\). A henselian field \((K, v)\) is called a tame field if \((\widetilde{K}|K, v)\) is a tame extension, where \(v\) still denotes the unique extension of \(v\) to the algebraic closure \(\widetilde{K}\) of \(K\). A root \(a\) of a polynomial \(f\in K[X]\) is a maximal root of \(f\) if for every root \(b\) of \(f\) we have \(v(x-a)\geq v(x-b)\). An extension \((K(x)|K, v)\) is weakly pure (in \(x\)) if \(x\) is a limit of a pseudo Cauchy sequence in \((K, v)\) of transcendental type, or there is \(a \in K\) such that \(v(x-a) = \max v(x-\widetilde{K})\). Let \(Q\in K[X]\) be a monic irreducible polynomial. Every element \(f(x)\) in \(K(x)\) can be written in a unique way as \(f(x)=f_0(x)+f_1(x)Q(x)+\cdots +f_j(x)Q(x)^j\), where \(f_0,f_1,\dots,f_j\) are polynomials of degrees less tha the degree of \(Q\). Then we let \(v_Q(f(x))\) be the minimum of \(v(f_0(x)),v(f_1(x)Q(x)),\dots,v(f_j(x)Q(x)^j)\).
In the following \((K,v)\) is a henselian valued field. The first theorem that the authors prove deals with a transcendental extension \((K(x)|K,v)\) such that there is a tame extension \((L'|K,v)\) where, for some extension of \(v\) from \(K(x)\) to \(L'(x)\), the extension \((L'(x)|L',v)\) is weakly pure (this involves the case where \((K,v)\) is a tame field). They denote by \(L\) the relative algebraic closure of \(K\) in a fixed henselization of \((K(x),v)\). They prove that the extension \((L(x)|L,v)\) is immediate and they construct a sequence of elements of \(L\) which is either finite or a pseudo Cauchy sequence of transcendental type which pseudo converges to \(x\). Every element of this sequence is a maximal root of its minimal polynomial over \(K\), and the sequence \((Q_{\nu})\) of these minimal polynomials is a complete sequence of key polynomials for \((K(x)|K,v)\) (i.e., for every \(f(x)\in K[x]\), \(v(f(x))\) is eventually equal to \(v_{Q_{\nu}}(f(x))\)). In the next theorem they prove similar results for a tame algebraic extension \((K(x)|K,v)\). Finally, they generalize these results to a countably generated tame extension \((L|K,v)\).