Find the greatest common divisor of $f(x)=2x^3+2x^2+x+4$ and $g(x)=x^4+3x^3+4x^2+3x$

Question

Find the greatest common divisor of $f(x)=2x^3+2x^2+x+4$ and $g(x)=x^4+3x^3+4x^2+3x$ in $\mathbb{Z}_{11}[x]$

Answer 1

We have to perform the Euclidean algorithm modulo $11$: $$\begin{align} h_1(x)&:=g(x)+(5x-1)f(x)\equiv 7x^2+7,\\ h_2(x)&:=f(x)+(6x+6)h_1(x)\equiv -x+2,\\ h_3(x)&:=h_1(x)+(7x+3)h_2(x)\equiv 2.\\ \end{align}$$ Hence, in $\mathbb{Z}_{11}[x]$, $$\gcd(g(x),f(x))=\gcd(h_1(x),f(x))=\gcd(h_1(x),h_2(x))=\gcd(h_3(x),h_2(x))=1.$$

Answer 2

You need to work in $$\mathbb{Z}_{11}[x]$$ where those fractions change to integers.

For example $(1/2) =6$ because $2\times 6 =1 $ in $\mathbb{Z}_{11}$

Also you need to continue the process further to get the g.c.d