What exactly is $k\left(T_{n}\right)_{n\in\mathbb{N}}$?

Question

Let $k$ be a field and $T_{n}$ indeterminates over $k$. Is $k\left(T_{n}\right)_{n\in\mathbb{N}}$ the field of fractions of the form $x=\frac{p}{q}$, where $p\in k\left[T_{i}\right]_{i\in\mathbb{N}}$ and $q\in k\left[T_{i}\right]_{i\in\mathbb{N}}\setminus\left\{ 0\right\} $ or is it something else, like a field of simple extensions?

Answer 1

Yes. It is the field of fractions of the polynomial ring $k[T_1,T_2,\dotsc]$ (which is an integral domain). It is in particular a field extension of $k$. When the adjoined elements are denoted by capital letters, one usually means variables. So $k(a,b)$ often means a field extension of $k$ in which $a,b$ may satisfy some relations, but $k(T_1,T_2)$ usually not.