Question

Let $f(x)\in\mathbb{Z}[x_{1},\dots,x_{n}]$ be a cubic homogeneous polynomial, which factors as $f(x)=g(x)h(x)$ over $\mathbb{C}$ with $\mathrm{deg}(g)=1$ and $h$ irreducible over $\mathbb{C}$. Assume that $n>3$. How can one prove that $\exists \lambda \in \mathbb{C}^{\times}$ such that $\lambda g \in \mathbb{Z}[x_{1},\dots,x_{n}]$?

I have seen this several times (for example the comment before Lemma 4.3 in this paper) but cannot really understand why. I feel that this can be proven by some Galois theory.

Answer 1

By unique factorization in $\mathbb C[x_1,\ldots,x_n]$, you have $g^\sigma=\mu g$ for each $\sigma\in\text{Aut}(\mathbb C)$ and $\mu$ depending on $\sigma$. Now pick $g$ such that one of its coefficients is $1$. Then $\mu=1$ for each $\sigma$. So all coefficients of $g$ are fixed under $\text{Aut}(\mathbb C)$, hence they are rational.