Let $K/\mathbb{Q}$ be a cyclic extension of degree $n$ with basis $(v_1,\ldots, v_n)$. Any $b\in K$ can be expressed by
$$b=x_1v_1+\dots+x_nv_n,$$ where $x_1,\ldots, x_n\in\mathbb{Q}$.
Do we know that
$$N_{K/\mathbb{Q}}(b)\in\mathbb{Z}[x_1,\ldots , x_n]?$$
Does it hold when $$N_{K/\mathbb{Q}}(b)=1?$$
It doesn't hold. Look at $\mathbb{Q} (i)$ with the basis $v_1=\frac{1}{2}+i$, $v_2=\frac{1}{3}+i$. For it, we have: $$w=x_1 v_1 + x_2 v_2 = (\frac{x_1}{2} + \frac{x_2}{3} ) + (x_1+x_2)i ,$$ $$N(w)=(\frac{x_1}{2} + \frac{x_2}{3} )^2+(x_1+x_2)^2 \not \in \mathbb{Z}[x_1,x_2] .$$