Showing the augmentation ideal of $\mathbb{F}[G]$ is a maximal ideal

Question

Given a basis of $\Bbb R^n,\ G:=\{e_0,...,e_{n-1}\}$, we define multiplication on the elements of the basis by $e_i\cdot e_j=e_{i+j}$ (where $i+j$ is calculated modulo $n$).

For a field $\Bbb F$ we define the ring $\Bbb F[G]=\{\sum_{j=0}^{n-1}a_je_j :a_j\in \Bbb F\}$ with the natural addition and multiplication.

We define a map $f:\Bbb F[G] \to \Bbb F$ by $f(\sum_{j=0}^{n-1}a_je_j)=\sum_{j=0}^{n-1}a_j$.

1. Prove this is a homomorphism
2. Prove that $\ker(f)$ is a maximal ideal.

What I did:

1. I succeeded showing it.

2. I think that $\ker(f)$ is only $0$, because otherwise the image would have been different than $0$. So actually I need to show that the only ideal in this field is $(0)$. I was wondering if this is a good direction?

Thanks.