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I have a question about the definition of a seed of a cluster algebra. It is said that a seed is a pair $(R, u)$, where $R$ is a quiver with $n$ vertices, $u = \{u_1, \ldots, u_n\}$ is a free generating set of the field $Q(x_1, \ldots, x_n)$, see Page 10 of the paper.

I think here $u_i$ is in terms of $x_1, \ldots, x_n$ and $u_1, \ldots, u_n$ generate $Q(x_1, \ldots, x_n)$ freely. Is this true? Thank you very much.

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    You have not yet make clear what it is you are trying to ask. I suggest you edit your question so that it ends with «What I want to to know is if SOMETHING».2012-08-15

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Each $u_i$ is a rational function in the $x_i$, with coefficients in $\mathbb{Q}$ - that's what it means to be an element of $\mathbb{Q}(x_1,\dotsc,x_n)$.

They set of $u_i$ also freely generates $\mathbb{Q}(x_1,\dotsc,x_n)$ as a field extension of $\mathbb{Q}$; another way of saying this is that every element of $\mathbb{Q}(x_1,\dotsc,x_n)$ can be written uniquely as a rational function in the $u_i$.

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    @AndreasBlass Corrected - I usually have $\mathbb{C}$ in place of $\mathbb{Q}$, which is where that came from.2013-08-29