# Finding a gcd in the form of a monic polynomial

The question is to find the greatest common divisor (in the form of monic polynomial) in $\Bbb F_5[X]$ of

$f=x^2-x+4$ and $g=x^3+2x^2+3x+2$

I used the Euclidian Algorithm for polynomials and found that the the GCD of these two polynomials is $2$.

How do I make this a monic polynomial?

In general rings or domains, gcds are defined only up to unit factors, since if $\,c\mid a,b\,$ then so too does $\,uc\,$ for $\,u\,$ any unit (invertible). In $\,\Bbb Z\,$ we can normalize gcds by choosing a nonnegative rep, i.e. multiply the gcd by $\,u=-1\,$ to make it positive if need be. Similarly, in your example, for polynomials over a field, we may normalize gcds by scaling them to be monic, i.e. scale the polynomial by the inverse of its leading coefficient to force the lead coefficient to be $\,1$.
• @jsan Yes, or equivalently, multiply by $\ \dfrac{1}2 \equiv \dfrac{6}2\equiv 3\pmod 5\ \$ – Bill Dubuque Mar 20 '14 at 17:23