Let 
be an integer polynomial. The 
can be factored into a product of two polynomials
of lower degree with rational coefficients iff it can be factored into a product of integer
polynomials of lower degree.
Gauss's Polynomial Theorem
See also
Abel's Irreducibility Theorem, Abel's Lemma, Gauss's Polynomial Identity, Kronecker's Polynomial Theorem, Polynomial, Schönemann's TheoremExplore with Wolfram|Alpha
References
Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 119, 1965.Referenced on Wolfram|Alpha
Gauss's Polynomial TheoremCite this as:
Weisstein, Eric W. "Gauss's Polynomial Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssPolynomialTheorem.html