Roots of generalized Schönemann polynomials in Henselian extension fields (Q999245)

Let \((F, v)\) be a valued field with residue class field \(\overline F,\) value group \(vF,\) and valuation ring \(A.\) One call a polynomial \(g \in A[X]\) a generalized Schönemann Polynomial over \((F, v)\) if it can be written in the form: \(g= p^e + t.h,\) where \(e\geq 1; p \in A[X]\) is monic with its canonical image \(\overline p\) in \(\overline F,\) is irreducible over \(\overline F;\) \(h \in A[X]\) has degree less than \(e.\deg(p);\) the polynomial \(\overline p\) does not divide \(\overline h;\) and finally, \(t \in A\) is nonzero and \(v(t) \notin svF\) for any divisor \(s>1\) of \(e.\) In the paper under review generalized Schönemann polynomials are studied over a valued field \(F.\) If such a polynomial \(g\) is tame (i.e., a root of \(g\) generates a tamely ramified extension of \(F\)). Moreover a criterion for when the existence in a Henselian extension field \(K\) of an approximate root of \(g\) guarantees the existence of an exact root of \(g\) in the extension field \(K,\) is given (see Theorem 5).