TST ALG3 Quiz2

Problem 4

Let $k$ be a field and let $R=k[x_1,\dots ,x_d]$ be a polynomial ring. Let $r,d⩾2$ and $f(x_1,\dots ,x_d)=x_1^r+x_2^r+\cdots +x_d^r$.

Exercise 341.1(Exercise).

Let $d=r=3$. Is $f$ irreducible? Does it depend on the characteristic of the field?