Factorization of polynomials: Difference between revisions

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===Factoring over '''Q''' and '''Z'''===

It can be shown that factoring over '''Q''' (the rational numbers) can be reduced to factoring over '''Z''' (the integers). This is a specific example of a more general case — factoring over a [[quotient field of fractions]] can be reduced to factoring over the corresponding [[integral ___domain]].

The classic proof, due to [[Carl Friedrich Gauss|Gauss]], first factors a polynomial into its ''content'', a rational number, and its ''primitive part'', a polynomial whose coefficients are pure integers and share no common divisor among them. Any polynomial with rational coefficients can be factored in this way, using a content composed of the greatest common divisor of the numerators, and the least common multiple of the denominators. This factorization is unique.