Definition 1

Let K be a commutative field and I a set. The field of fractions (I, p. 116) of the integral domain $K[(X_i)_{i \in I}]$ is denoted by $K((X_i)_{i \in I})$ or $K(X_i)_{i \in I}$. Its elements are called rational fractions in the indeterminates $X_i$ with coefficients in K.

For $I = \{1, 2, \ldots, n\}$ we write $K(X_1, X_2, \ldots, X_n)$ in place of $K((X_i)_{i \in I})$.

Let A be an integral domain and K its field of fractions. The ring $A[(X_i)_{i \in I}]$ may be identified with a subring of $K[(X_i)_{i \in I}]$, hence also of $K((X_i)_{i \in I})$. For each $f \in K[(X_i)_{i \in I}]$ there exists a non-zero element $\alpha$ of A such that $\alpha f \in A[(X_i)_{i \in I}]$. Hence every element of $K((X_i)_{i \in I})$ can be written as $u/v$ where $u, v \in A[(X_i)_{i \in I}], v \neq 0$. Thus $K((X_i)_{i \in I})$ may be identified with the field of fractions of $A[(X_i)_{i \in I}]$.

Now let K be a commutative field, I a set and J ⊂ I. Put B = K[(X_i)_{i ∈ J}], then K[(X_i)_{i ∈ I}] = B[(X_i)_{i ∈ I - J}], and by what has been said, K((X_i)_{i ∈ I}) may be identified with K'((X_i)_{i ∈ I - J}), where K' = K((X_i)_{i ∈ J}).