On the irreducibility of a class of polynomials, III

AbstractThis work is a continuation and extension of our earlier articles on irreducible polynomials. We investigate the irreducibility of polynomials of the form g(f(x)) over an arbitrary but fixed totally real algebraic number field L, where g(x) and f(x) are monic polynomials with integer coefficients in L, g is irreducible over L and its splitting field is a totally imaginary quadratic extension of a totally real number field. A consequence of our main result is as follows. If g is fixed then, apart from certain exceptions f of bounded degree, g(f(x)) is irreducible over L for all f having distinct roots in a given totally real number field