Two questions from ring homomorphisms.

Question

1. What is natural homomorphism $f\colon \Bbb C[x]\to \Bbb C[x,x^{-1}]$? Is it just embedding?

2. Are all prime ideals in $\Bbb C[x,x^{-1}]$of the form $\{\,g: g(y)=0\,\}$ for chosen points $y$?? Of course, sets of its elements which take value $0$ in chosen point are prime and maximal ideals, but are there more ideals or no? If natural homomorphism is embedding then there shouldn't be more ideals because counterimage of any prime ideal in $\Bbb C[x,x^{-1}]$ is prime in $\Bbb C[x]$, and embedding is 1-1. Am I right?