Consider the polynomial ring $Q[x]$. Let $p(x) = x^3-2$. Let $I$ be an ideal generated by $p(x)$. Show that each element of $\frac{Q[x]}{I}$ is of the form $a_0+a_1t+a_2t^2$ with $a_0, a_1, a_2$in $Q$ and $t = x+I$
My approach: $I$ is a maximal ideal since $p(x)$ is irreducible. So $\frac{Q[x]}{I}$ is a field. Please help me.
I couldn't proceed from here.
I am doing a graduation course in Ring theory.
Note that $\Bbb Q $ is a field $\implies \Bbb Q[x]$ is a PID $\implies \Bbb Q[x]$ is an Euclidean Domain.
Hence the Division Algorithm holds in $\Bbb Q[x]$.
So we can write for any $f(x)$ and $g(x)\neq 0$ ; $f(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or $\deg r(x)<\deg g(x)$
Consider $f(x)\in \Bbb Q[x]/I$ and take $g(x)=p(x)$ can you find the elements of the quotient ring now?