I'm trying to show that $\frac{\mathbb{C}[X]}{<x-1>} \cong \mathbb{C}$ and I am not sure if this argument is correct.
Define $\phi: \mathbb{C}[X] \to \mathbb{C}$ by $\sum a_it^i \to \sum a_i$. My aim is to show that $ker\phi = <x-1>$. Now we know by the first isomorphism theorem that the image of this map is a subset of $\mathbb{C}$ and hence it is an integral domain, and so $ker \phi$ must be prime, and as $\mathbb{C}[X]$ is a P.I.D then $ker\phi$ must be generated by an irreducible polynomial. Now $(x-1)$ is an irreducible polynomial in the kernel, and so $ker\phi = <x-1>$
Does this argument work?
Thanks