In this article, the author tells that let $F$ be a field with $\mathrm{char}F\neq2$ and $L=F(\ldots,x_{-1},x_0,x_1,\ldots)$ be the field extension by adjoining a countable set of indeterminates $(x_n)_{n=-\infty}^{+\infty}$ to $F$.
I would like to ask what the form of its elements is. I think that $L$ is the field generated by $(x_n)_{n=-\infty}^{+\infty}$ over $F$. However, recall that the field generated by a set $S$ over a field $F$ is defined as follows.
Let $F$ be a field and let $K$ be an extension field of $F$ and $S$ a subset of $K$. The intersection of all subfields of $K$ containing both $F$ and $S$ is called the field generated by $S$ over $F$, denoted by $F(S)$.
However, I am unknown what here $K$ in that article is. Please help me explain this one. Any counterexample or reference or technique is very much appreciated. Thank you in advance.