On reducible trinomials. III
In general, to decide if a given polynomial is irreducible or not is a difficult problem, however, several results have been established for families of polynomials. Among those results, one finds Capelli's theorem, see \textit{A. Schinzel} [Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications 77, Cambridge University Press (2000; Zbl 0956.12001), Theorem 19, p. 92], which characterizes irreducible binomials over a field. For some trinomials the situation is as follows. Let \(K\) be a field of characteristic \(p\), \(\bar{K}\) an algebraic closure of \(K\), \(K({\mathbf y})\) the rational function field in \(n\) variables \({\mathbf y}=(y,\ldots,y)\), \(L\) a finite separable extension of \(K(\mathbf {y})\) and \(T(x;A,B)=x^n+Ax^m+B \in K({\mathbf y})[x]\), where \(n>m>0\), \(p\) does not divide \(mn(n-m)\) and either \(A,B\in K({\mathbf y})^*\), \(A^{-n}B^{n-m}\not \in K\) or \(A,B\in L^*\), \(A^{-n}B^{n-m}\not\in \bar{K}\). Set \(d=\gcd(n,m)\), \(n_1=n/d\) and \(m_1=m/d\) and \(g(x)=x^{n_1}+Ax^{m_1}+B\), then \(T(x;A,B)=g(x^d)\). By Theorem 25, p. 122 of the author's above book, a criterion is known that ensures the reducibility of \( T(x;A,B)\). This criterion is related to the existence of a linear or quadratic factor \(F(x)\) of \(g(x)\). In the present paper, the author studies the reducibility of the polynomial \( T(x;A,B)F(x^d)^{-1}\). To get a wider account on irreducibility of trinomials, see the author's works [Diss. Math. 329, 83 p. (1993; Zbl 0795.12003); Errata, Acta Arith. 73, 399-400 (1995; Zbl 0839.12001); and Publ. Math. 56, 575-608 (2000; Zbl 0960.12001)].
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