Invertible elements of $K[x]/\langle p(x)\rangle$ when $p(x)$ is irreducible

Question

I want to find the inverse of $f(x)=ax^{2}+bx+c$ in $L=K[x]/\langle p(x)\rangle$ when $p(x)$ is an irreducible polynomial in $K[x]$ with degree $3$.

I know elements of $L$ are like $f(x)$ such that $a,b,c$ are in $K$. I'm choosing element $g(x)=a'x^{2}+b'x+c'$ in $L$ and seting, $f(x)g(x)=1$ and know $c'$ is inverse of $c$, but finding $a'$ and $b'$ not simple because I think finding these depend on knowing $p(x)$! Namely, $f(x)g(x)$ must compute $(\!\!\!\mod\!\! p(x))$ to place $x^{4}$ and $x^{3}$ with power less than these.

Please help me to find $g(x)$. Thank you.

Answer 1

Without knowing $p(x)$ and $f(x)$ can not solving this problem; and note, degree of $g(x)$ can be $2$. for example, if $K=\mathbb{Q}$, $p(x)=x^{3}-2x-2$ and $f(x)=x^{2}+x+1$ then $g(x)=-\dfrac{2}{3}x^{2}+\dfrac{1}{3}x+\dfrac{5}{3}$ such that $fg=1$ in $K[x]/\langle‎p(x)\rangle‎$.