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!
2026-05-16 10:38:15.1778927895
$T_1^{n+1}-T_2^{n}$ is prime for $n\in\mathbb{N}$.
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Hint: Consider the principal ideal generated by $T_1^{n+1} - T_2^n$ and show that this is a prime ideal by looking at the quotient $k[T_1,T_2]/(T_1^{n+1} - T_2^n)$. Can you maybe find an integral domain that is isomorphic to $k[T_1,T_2]/(T_1^{n+1} - T_2^n)$?