Abstract
In this paper, all irreducible factors of bivariate polynomials of the form f(x) - g(y) over an arbitrary field are described. It also proves that the number of irreducible factors of f(x) - g(y) (counting multiplicities) does not exceed the greatest common divisor of the degrees of f(x) and g(y), yielding a well known result of Tverberg regarding the irreducibility of f(x) - g(y). It proves that if f(x) and g(y) are non-constant polynomials with coefficients in the field ℚ of rational numbers and deg f(x) is a prime number, then f(x) - g(y) is a product of at most two irreducible polynomials over ℚ. This contributes to a problem raised by Cassels which asks for the polynomials f, such that the polynomial [Formula: see text] is reducible.
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