What are function fields?

Question

Given a field $F$, I know that $F[X]$ is the ring of polynomials in $X$. I know that this is not a field.

I have seen the notation for $F(x)$ with round brackets. Usually when we use round brackets we take the smallest field containing $F$ and $x$. For example, if $F(\alpha) = F[\alpha]$ for all algebraic $\alpha \in E$ ($E$ some extension of $F$).

My question is what $F(x)$ is when $x$ is a variable. Is this just the quotient of polynomials? What is the definition? Is this what is called a function field.

Answer 1

$F(x)$ is:

• The smallest field containing $F$ and the variable $x$;

• The set of symbolic quotients $\frac{p(x)}{q(x)}$, where $p,q\in F[x]$ and $q\neq 0$.

• The field of fractions associated to the ring $F[x]$.

Answer 2

Yes, you can define $F(X)$ as the quotient field of the ring of polynomials $F(X)$; informally, all rational expressions in $X$.

It is indeed an example of an (algebraic) function field in one variable (over $F$). But there are other (algebraic) function field in one variable (over $F$) too.