Polynomial greatest common divisor

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In algebra, the greatest common divisor (frequently abbreviated GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field, the polynomial GCD may be computed as for the integer GCD, with the Euclidean algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant. The similarity between integer GCD and polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this provides information on the roots without computing them. For...

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