Describe all ring homomorphisms from $\mathbb R[t] /\langle t^4+t^3-3t-3, t^6-9\rangle$ to $\mathbb C$.

Question

Describe all ring homomorphisms from $\mathbb R[t] /\langle t^4+t^3-3t-3, t^6-9\rangle$ to $\mathbb C$.

What determines the homomorphisms? This seems quite confusing. Any general explanation on how to solve this type of problems would be much appriciated!

Answer 1

Hint: If $K$ is a commutative ring and $f_1,\dotsc,f_n \in K[x]$, then $K$-algebra homomorphisms $K[x]/(f_1,\dotsc,f_n) \to A$ correspond to elements $a \in A$ such that $f_1(a)=\dotsc=f_n(a)=0$.

As remarked by Jyrki, your task is probably to determine the $\mathbb{R}$-algebra homomorphisms (not all ring homomorphisms).

Thus, you only have to solve the equation system $t^4+t^3-3t-3=t^6-9=0$ in $\mathbb{C}$.