A resultant condition for the irreducibility of the polynomials
Main result (theorem 1): Let (K,v) be a discrete valued field, A the valuation ring of v, \(\bar K\) its residue field, p in K such that \(v(p)=1\) and X an indeterminate over K. Let F be a monic polynomial in A[X] such that \(F=f_ 1^{m_ 1}...f_ n^{m_ n}+pg\) where \(f_ 1,...,f_ n\) and g are in \(A[X]\), \(f_ 1,...,f_ n\) are monic, \(\bar f{}_ 1,...,\bar f_ n\) are nonconstant and irreducible in \(\bar K[X]\) and \(\bar f_ i\) does not divide \(\bar g\) if \(m_ i\) is not less then 2. Let \(h=f_ 1^{m_ 1}...f_ n^{m_ n}.\) If, for every nontrivial factorization \(h=h_ 1h_ 2\), there is \(j=1,2\) and \(i=1,...,n\) such that \(f_ i\) divides \(h_ j\) and for all divisors d of \(R(f_ i,g)\), \(R(\bar f_ i,\overline{h/h_ j})\neq \bar d\) then F is irreducible in K[X]. From this result the criteria for irreducibility of Schönemann, Eisenstein and Akira are derived.
scientific article; zbMATH DE number 3995842