I'm reading Complex Analytic and Differential Geometry by J-P. Demailly. In section 2.4.2, the author gives the following proposition:
Suppose $A=V(\mathcal{J})$, where $J$ is a prime ideal of $\mathcal{O}_n=\mathcal{O}_{\mathbb{C}^n,0}$. Set $\mathcal{J}_k=\mathcal{J}\cap \mathbb{C}\{z_1,\cdots,z_k\}$.
Proposition. There exist an integer $d$, a basis $\left(e_1, \ldots, e_n\right)$ of $\mathbb{C}^n$ and associated coordinates $\left(z_1, \ldots, z_n\right)$ with the following properties: $\mathcal{F}_d=\{0\}$ and for every integer $k=d+1, \ldots, n$ there is a Weierstrass polynomial $P_k \in \mathscr{J}_k$ of the form $$ P_k\left(z^{\prime}, z_k\right)=z_k^{s_k}+\sum_{1 \leqslant j \leqslant s_k} a_{j, k}\left(z^{\prime}\right) z_k^{s_k-j}, \quad a_{j, k}\left(z^{\prime}\right) \in \mathscr{J}_{k-1}, $$ where $a_{j, k}\left(z^{\prime}\right)=O\left(\left|z^{\prime}\right|^j\right)$. Moreover, the basis $\left(e_1, \ldots, e_n\right)$ can be chosen arbitrarily close to any preassigned basis $\left(e_1^0, \ldots, e_n^0\right)$.
Therefore we have an injective ring morphism $\mathcal{O}_d=\mathbb{C}\{z_1,\cdots,z_d\}\hookrightarrow \mathcal{O}_n/\mathcal{J}$. And the ring extension is a finite integral extension. Thus set $M_A$ and $M_d$ the quotient fields of $\mathscr{O}_n / \mathcal{F}$ and $\mathscr{O}_d$ respectively. Then $M_A=$ $\mu_d\left[\tilde{z}_{d+1}, \ldots, \tilde{z}_n\right]$ is a finite algebraic extension of $M_d$. Let $q=\left[M_A: M_d\right]$ be its degree and let $\sigma_1, \ldots, \sigma_q$ be the embeddings of $M_A$ over $M_d$ in an algebraic closure $\bar{M}_A$. By the primitive element theorem, there exists a linear form $u\left(z^{\prime \prime}\right)=c_{d+1} z_{d+1}+\cdots+c_n z_n, c_k \in \mathbb{C}$, such that $M_A=M_d[\tilde{u}]$.
My question is : by primitive element theorem we get $M_A/M_d$ is a simple extension, so $u\in M_A$ can be written as $u\left(z^{\prime \prime}\right)=c_{d+1} z_{d+1}+\cdots+c_n z_n, c_k \in M_d$. By why can he claim that we can select $u$ such that $c_k\in \mathbb{C}$? I'm sorry for such a long question. Thanks in advance.