Let $\mathbb{C}[X_1,...,X_d]$ denote the set of polynomials over $\mathbb{C}$ in $d$ variables. If $d=1$, we know that the irreducible polynomials are exactly the polynomials of degree $1$, i.e. linear polynomials. Is there any similar characterization of the irreducible polynomials of $\mathbb{C}[X_1,...,X_d]$?
It seems to be a bit harder, since for example the polynomial $X_1^2 + X_2^2$ should also be irreducible.