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

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

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

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

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