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Gauss's Polynomial Theorem

If a Polynomial


with integral Coefficients is divisible into a product of two Polynomials $f=\psi\phi$
$\displaystyle \psi$ $\textstyle =$ $\displaystyle x^m+\alpha_1x^{m-1}+\ldots+\alpha_m$  
$\displaystyle \phi$ $\textstyle =$ $\displaystyle x^n+\beta_1 x^{n-1}+\ldots+\beta_n,$  

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


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

© 1996-9 Eric W. Weisstein