Let $\mathbb{K}$ be a field. Then $ \mathbb{K}[X,Y] /(XY- 1)$ has infinitely many prime ideals.[True/False].
What happens when $\mathbb{K}= \mathbb{C}?$
What I know is that, the prime ideals in $\mathbb{K}[X,Y] /(XY- 1)$ are in one-one correspondence with the prime ideals in $\mathbb{K}[X,Y] $ containing the ideal $(XY-1).$
Can we use the result here? Any help would be appreciated. Thanks in advance.
Note that $\mathbb{K}[X,Y]/(XY-1) \cong \mathbb{K}[X]_X$. This last ring is the one obtained by inverting $X$ in $\mathbb{K}[X]$. Thus, the prime ideals of $\mathbb{K}[X,Y]/(XY-1)$ are in bijection with those of $\mathbb{K}[X]_X$. Moreover, the prime ideals of $\mathbb{K}[X]_X$ are in bijection with those of $\mathbb{K}[X]$ that do not contain $X$. Now, the only prime ideal of $\mathbb{K}[X]$ that contains $X$ is $(X)$. (This is because $(X)$ is in fact maximal.)
We conclude that, if $\mathbb{K}[X]$ has infinitely many prime ideals, then $\mathbb{K}[X,Y]/(XY-1)$ also has infinitely many prime ideals.