Construct a bijection $\mathrm{Hom}_{\mathbb C}(\mathbb C[x,y]/(xy-1),\mathbb C) \to \mathbb C\setminus \{0\}$

Question

The question is :

Construct a bijection $\mathrm{Hom}_{\mathbb C}(\mathbb C[x,y]/(xy-1),\mathbb C) \to \mathbb C\setminus \{0\}$.

Here $\text{Hom}_{\mathbb C}(\mathbb C[x,y]/(xy-1),\mathbb C)$ is the set of all homomorphisms $$\phi :\mathbb C[x,y]/(xy-1) \to \mathbb C\;\; \text{s.t.}\;\; \phi|_{\mathbb C}=\mathrm{id}|_{\mathbb C}.$$

Please someone give some hints how can I do this?

Thank you.

Answer 1

A homomorphism from $R=\Bbb C[x,y]/(xy-1)$ to $\Bbb C$ is a homomorphism $\Phi$ from $\Bbb C[x,y]$ to $\Bbb C$ with $\Phi(xy-1)=0$. Each homomorphism $\Phi:\Bbb C[x,y]\to \Bbb C$ has has the form $\Phi_{a,b}:f(x,y)\to f(a,b)$ where $a$, $b\in\Bbb C$. Then $\Phi_{a,b}(xy-1)=ab-1$. So $\Phi_{a,b}$ defines a homomorphism on $R$ iff $ab=1$, etc.

Answer 2

Hint: $~$ A $\mathbb{C}$-algebra homomorphism $\mathbb{C}[x,x^{-1}]\to\mathbb{C}$ is determined by where $x$ is sent.