[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"]
"Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = \alpha_1x_1 +\dots+\alpha_nx_n$ with $\alpha_j \in K$. Suppose that $\alpha_1,\dots,\alpha_n$ are linearly independent over $\mathbb{Q}$. Then $n\le d$ and $L$ will not vanish on $\mathbb{Q}^n \backslash 0$. As usual, denote the embeddings of $K$ into $\mathbb{C}$ by $\alpha \to \alpha^{(i)}, (i=1,\dots,d)$ and write $L^{(i)}(X) = \alpha_1^{(i)}x_1+\dots+\alpha_n^{(i)}x_n$. We write
$$N(L(X)) = \prod_{i=1}^d L^{(i)}(X).$$
A norm form is any form $F$ of the type $F(X) = cN(L(X))$ for some $L$ as above and $c \in Q^{*}$. For a norm form $F$, we have $F(X) \in \mathbb{Q}[X]$."
Question:
Why do we have $F(X) \in \mathbb{Q}[X]$?
For fixed $j$, I can see that $\prod_{i=1}^{d}\alpha_j^{(i)} \in \mathbb{Q}$. But what happens when we have coefficients in $F(X)$ of the form $\alpha_{j_1}^{(1)}\alpha_{j_2}^{(2)}\dots\alpha_{j_d}^{(d)}$ where some of the $j_k$'s are different?
Thank you.
The coefficients in the norm form are invariant under the action of the Galois group, so they are in the ground field.