Ancillary Results About Polynomials

Definition 1   Let $ f_1$ and $ f_2$ be polynomials. If $ f_1$ and $ f_2$ have no non-trivial common factor, then they are said to be coprime. That is, $ f_1$ and $ f_2$ are coprime if $ f$ is a constant polynomial whenever $ f \vert f_1$ and $ f \vert f_2$.

Theorem 1 (Euclid's algorithm)   Let $ a$, $ b$ be polynomials. Then there exist polynomials $ q$, $ r$ with $ r=0$ or $ \partial r < \partial b$ such that

$\displaystyle a(x) = q(x) b(x) + r(x).

Euclid's algorithm is really just long division. We are applying it here to polynomials, where $ q(x)$ is the `quotient' and $ r(x)$ is the `remainder'.

Theorem 2   If $ f_1$ and $ f_2$ are coprime polynomials, then there exists polynomials $ a_1$ and $ a_2$ such that

$\displaystyle 1 = a_1(x) f_1(x) + a_2(x) f_2(x).

This is a corollary of Euclid's algorithm, and is equivalent to the definition of coprimality.

Gihan Marasingha 2005-09-19