Question about showing that $\mathbb{K}[X,Y]/(XY-1)$ is isomorphic to $\mathbb{K}[T,T^{-1}]$.

Question

I want to show that $\mathbb{K}[X,Y]/(XY-1) \cong \mathbb{K}[T,T^{-1}]$ and I’ve asked myself it is possible to show this by using the euclidean division of a polynomial in $\mathbb{K}[X,Y]/(XY-1)$?

Answer 1

The isomorphism is $X \to T$, $Y \to T^{-1}$. Just try it out the details are not hard. The intuition is that in $\mathbb{K}[X,Y]/(XY-1)$, the relation $XY = 1$ means that $Y$ is now the multiplicative inverse of $X$ (so, serves the same purpose as $X^{-1}$).