Find polynomials $m,n \in \mathbb{F}_5[X]$ such that $mf + ng = h$.

Question

Let $f, g \in \mathbb{F}_5[X]$ where $f=X^9+X^8+ \cdots +X^2+X+1, g=X^4+X-2$.

(1) Find the greatest common divisor $h$ of $f$ and $g$.

(2) Find polynomials $m,n \in \mathbb{F}_5[X]$ such that $mf + ng = h$.

(1) By using polynomial division, I found out that $h = 3X^2+4X+3$.

I don't know how to approach (2) (systematically), so could somebody please show me how that would work?

Thanks in advance!

Answer 1

Approach 2 by using Euclid's algorithm in reverse (use your answer to part 1).

i.e. Take your answer h = gcd(f,g), rewrite the terms on the gcd side you have until you work back to both f and g. Keep rewriting terms until you get h = mf + ng.