Suppose we have an algebraically closed field $F$ and $n+1$ variables $X_0, \dots, X_n$, where $n > 1$. Does there exist an irreducible homogeneous polynomial in these variables of degree $d$ for any positive integer $d > 1$? In other words, does there always exist an irreducible hypersurface of arbitrary degree?
Of course, I am also interested in constructions of these polynomials.
Thank you.
Best Answer
Yes, this is straightforward. First note that a homogeneous polynomial $f(X_0, ... X_n)$ which is not divisible by $X_0$ is irreducible if and only if $f(1, ... X_n)$ is irreducible, so the problem reduces to constructing irreducible polynomials in $k[x_1, ... x_n]$ of degree $d$. To do this we can take $x_1^2 - x_2 h(x_3, ... x_n)$ for any polynomial $h$ of degree $d-1$. (If $n = 2$ then instead use, for example, $x_1^2 - (x_2 + x_2^d)$. The important thing is that whatever comes after $x_1^2$ should not itself be a square.)