Gauss's Polynomial Theorem

If a Polynomial

$$f(x)=x^N+C_1x^{N-1}+C_2x^{N-2}+\ldots+C_N$$

with integral Coefficients is divisible into a product of two Polynomials $f=\psi\phi$

$$\psi = x^m+\alpha_1x^{m-1}+\ldots+\alpha_m$$

$$\phi = x^n+\beta_1 x^{n-1}+\ldots+\beta_n,$$

then the Coefficients of this Polynomial are Integers.

See also Abel's Irreducibility Theorem, Abel's Lemma, Kronecker's Polynomial Theorem, Polynomial, Schoenemann's Theorem

References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 119, 1965.