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

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.

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 $$1$$.