$T_1^{n+1}-T_2^{n}$ is prime for $n\in\mathbb{N}$.

Question

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!

It is prime (even irreducible) since $gcd(n+1,n)=1$, see the duplicate. — Dietrich Burde, Apr 02 '20 at 13:59

score 1 · Accepted Answer · answered Apr 02 '20 at 13:51

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 $k[T_1,T_2]/(T_1^{n+1} - T_2^n)$?

$T_1^{n+1}-T_2^{n}$ is prime for $n\in\mathbb{N}$.

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