finite dimensional integral domain containg $\mathbb C$ Let $ R$ be an integral domain containing $\mathbb C$.
Suppose that $R$ is a finite dimensional $\mathbb C$-vector space . Show that $R=\mathbb C$.
One side $\mathbb C \subset R$ is obvious. What about the other?
Show me right way.Thanks in advance.

Let $z\in R$. define the application $f_z$ on $R$ by $f_z(x)=xz$. This is a $\mathbb{C}$-linear operator on the finite dimensional $\mathbb{C}$-vector space $R$. Hence it has an eigenvalue $\lambda\in \mathbb{C}$. So there exists $y\in R$, non zero, such that $f_z(y)=\lambda y=yz$. This imply $(z-\lambda)y=0$, and as $R$ is integral, we get $z=\lambda\in \mathbb{C}$. 
