$K[x+y] \subseteq K[x, y]/(xy-1)$ is an integral extension

Question

Let $K$ be a field. Prove that $K[x+y] \subseteq K[x, y]/(xy-1)$ is an integral extension.

I know that $K[x,y]/(xy-1) \simeq K[t, t^{-1}]$, but I'm not sure if this would be useful to prove the statement.

Answer 1

You can write $K[x,y]/(xy-1) \simeq K[t,t^{-1}]$ (I'd say it is useful to see what is the algebra and so one doesn't bother with the quotient), then $K[x+y] \simeq K[t+t^{-1}]$, and it suffices to show that $t$ and $t^{-1}$ are roots of polynomials with coefficients in $K[t+t^{-1}]$, the leading coefficients being $1$.