Can someone give me an idea, how to prove that $\mathbb{Q}[x]/I=\mathbb{Q}[\hat{x}]$, where $I$ is the ideal generated by $-1+x+x^{2}$ and $\hat{x}$ is the equivalence of $x$ in $\mathbb{Q}[x]/I$ ?
I know some facts about $\mathbb{Q}[x]/I$ (it is a field, since the polynomial is irreducible etc.), but my problem is that we did so much theory in class the I literally can't see the forest for the trees. I tried applying the proof from theorem 3 from page 512 from Dummit \& Foote's {}``Abstract Algebra'', but somehow I couldn't get it to work, because my hunch to solve this problem was to use the evaluation homomorphism, but plugging an equivalance class of polynomials into a polynomial itself is just extremely confusing.