On irreducible polynomials in several variables which become reducible when the variables are replaced by powers of themselves
Eli Gourin · Transactions of the American Mathematical Society · 1930
ELI GOURIN Let Q(xx, •••,*,) be a polynomial in Xi, • • • , xs, irreducible in the field of all constants.There exist, in certain cases, integers ti, • • • , t, such that the polynomial Q(xitl, • • • , x,1') is reducible.The determination of the integers t for which reducibility occurs is a problem which arose in an investigation of J. F. Ritt on the factorization of exponential forms.fThe case of real interest is that in which Q has at least three terms.We shall, in this introduction, limit ourselves to the discussion of the results for this case.The relatively simple case of two terms * Presented to the Society, February 23,1929; received by the editors in February, 1930.t A factorization theory for funclionsYil-iaie"^, these Transactions, vol.29 (1927), pp.584-596.X That is,if in any irreducible factor of Q(x, Hi, ■ ■ ■, x3'i>) each x¡¡ is replaced by xkak the resulting polynomial will be irreducible.485 * Assuming the existence of these sets, it is easy to conclude that no infinite system of distinct sets satisfying the conditions (1) and (2) of the theorem can exist.See also §6.