I'm reading Artin's "Galois Theory", here:
A. Extension Fields.
If $E$ is a field and $F$ a subset of $E$ which, under the operations of addition and multiplication in $E$, itself forms a field, that is, if $F$ is a subfield of $E$, then we shall call $E$ an extension of $F$. The relation of being an extension of $F$ will be briefly designated by $F\subset E$. If $\alpha,\beta,\gamma,\dots$ are elements of $E$, then by $F(\alpha,\beta,\gamma,\dots)$ we shall mean the set of elements in $E$ which can be expressed as quotients of polynomials in $\alpha,\beta,\gamma,\dots$ with coefficients in $F$. It is clear that $F(\alpha,\beta,\gamma,\dots)$ is a field and is the smallest extension of $F$ which contains the elements $\alpha,\beta,\gamma,\dots$. We shall call $F(\alpha,\beta,\gamma,\dots)$ the field obtained after the adjunction of the elements $\alpha,\beta,\gamma,\dots$ to $F$, or the field generated out of $F$ by the elements $\alpha,\beta,\gamma,\dots$. In the sequel all fields will be assumed commutative.
What does he means by "quotient of polynomials in $\alpha,\beta,\gamma,\dots?$
By "quotients of polynomials" Artin means rational functions of $\alpha,\beta,\gamma\dots$ like $$\frac{c_1\alpha}{c_2\beta+c_3\gamma}$$ where the $c_i$ are elements of the smaller field $F$.