Let $P$ be a free multiplicative abelian group of a finite rank $m$ with generators $g_1,\dots,g_m$. Field $F$ of rational functions in $n$ independent variables is set to be the ambient field in our case with the coefficient in the field of fractions of the integer group ring $\mathbb{Z}P =\mathbb{Z}[g_{1}^{\pm1},\dots,g_{m}^{\pm1}]$. Next, we have a definition of a seed:
A seed in $F$ is a pair $\sum=$(x,$\tilde{B}$) where x$=(x_1,\dots,x_n)$ is a transcendence basis of $F$ over the field of fractions of $\mathbb{Z}P$ and $\tilde{B}$ is an $ n \times (n+m)$ integer matrix whose principal part is sign-skew-symmetric
In what follows x is called a cluster, and its elements $x_1,\dots ,x_n$ are calledd cluster variables. What I am not sure about is how do we know that x is a transcendence basis of $F$ over the field of fractions of $\mathbb{Z}P$? So I think that $\mathbb{Z}P$ $\subset F$(?). So I first need to show that x is algebraically independent over $\mathbb{Z}P$? And that the extension $\mathbb{Z}P(x_i;i\in \{ 1, \dots, n\}) \subset F$ is algebraic (here, field of fractions of a polynomial ring $k[x_i;i\in I]$ is denoted $k(x_i;i\in I)$ for some field $k$)? Is that correct? If so, I am not really sure how do this. I would appreciate some hints.