If $F$ is a field, what does the notation $F(x)$ mean? If $F$ is a field, what does the notation $F(x)$ mean? I am trying to understand transcendence degree of field extension, and I am stuck in this notation. 
More context: I am reading this pdf, and my problem is in the very first page. 
 A: In the context of field extensions (as you mentioned in the question), $F(x)$ is the smallest possible field containing $F$ and $x$. For example, one can construct $\mathbb{Q}(\sqrt{2})$ by adjoining the numbers $a+b\sqrt{2}$ to $\mathbb{Q}$ where $a,b$ are arbitrary rational numbers. Note that this does result in a field, because sums, products, and reciprocals of numbers of this form are again numbers of this form.
A: If I am not mistaken, both Ian and ihf are right, depending on what the $x$ is in $F(x)$. If $x$ is an indeterminate, then the smallest field containing both $F$ and $x$ is the field of fractions of the polynomial ring $F[x]$. If $\alpha$ is the root of a polynomial in $F[x]$, as is the case with $\sqrt{2}$ in $\mathbb{Q}[x]$ then $F(\alpha)$ will be the field $F[x]/(p(x))$ where $p(x)$ is the monic irreducible polynomial in $F[x]$ with $p(\alpha)=0$.
It would be considered awkward notation to use $x$ to represent the second case(which is why I used $\alpha$ to suggest it is not an indeterminate.), but not technically wrong. It would be my guess that if you see $F(x)$ without any other qualifiers, they are assuming $x$ is an indeterminate.
A: Since you mention transcendence degree, $F(x)$ is probably the field of rational functions in one variable, that is, the field of fractions of the polynomial ring $F[x]$.
