let $\mathbb{K}$ be an algebraically closed field. Does anyone have an idea how to show that for every $n\in\mathbb{N}$ $$T_1^{n+1}-T_2^{n}\in\mathbb{K}[T_1,T_2]$$ is prime?   Thank you very much!
I am trying to solve the following exercise.      Assume that $m$ and $n$ are co-prime positive integers, and that $k$ is a field. Prove that the ideal generated by $x^{m} - {y^n}$ is a prime ideal in $k[x,y]$.   I tried the following approach : Consider the ring homomorphism ${\varphi}:k[x,y]\rightarrow{k[t]}$ defined by $\varphi(x) = t^n$, and $\varphi(y) = t^m$, and extend $\varphi$ to $k[x,y]$ to ensure that it is a ring homomorphism. Then the ideal $(x^m - y^n)$ is clearly contained in the kernel of $\varphi$. It remains to prove the reverse set inclusion. i.e. to prove that this ideal is exactly equal to the kernel of $\varphi$. I don't know how to go about this!
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