bijection between points and algebra homomorphisms

Question

if we consider $\psi:V\to Hom(\mathbb{F}[V],\mathbb{F})$ such that $a\mapsto ev_a$. Is this map a bijection when $\mathbb{F}$ is algebraically closed?**

Answer 1

But is this map the unique F-algebra homomorphism with this kernel?

Yes. If you have another map $\mathbb{F}[V] \rightarrow A$ with the same kernel, there exists a unique isomorphism between the images, commuting with the maps. This is a corollary of the homomorphism theorem for rings.

Answer 2

The maps will be the same up to isomorphism.

For your second question, remember: for a$\in$ V to be in the kernel, f(a)=0 in $\mathbb{F}[V]$,the quotient ring.