Is norm form a polynomial with integer coefficients?

Question

Let $k$ be a number field of degree $n$ over $\mathbb{Q}$. Pick an integral basis $w_1, ..., w_n$. Then we can define $$ N(x_1, .., x_n) = N_{k/\mathbb{Q}}(x_1 w_1 + ... + x_n w_n). $$ I was wondering how we can show that this is a degree $n$ homogeneous form. And are the coefficients necessarily in $\mathbb{Z}$?