Examples of field of fractions

Question

I know that the field of fractions of $\mathbb{Z}$ is $\mathbb{Q}$. I also know that the field of fractions of $F[x]$ ($F$ field) is $F(x)$.

After searching around I haven't been able to find any other simple examples of this. Maybe the problem is that I don't know a lot of examples of integral domains.

I would like to see other examples of fields of fractions.

Answer 1

The field of fractions of the ring of holomorphic functions on a certain domain $\Omega$ (open and connected in $\mathbb{C}$) is the field of meromorphic functions on $\Omega$.

The field of fractions of the ring of formal power series over the field $F$ is the field of formal Laurent series.

Answer 2

There is an infinite chain of fields you can make using the field of fractions starting from any given field as follows. Take a field, like $\mathbb{Z_{p}}$, and form the polynomial ring in the indeterminate $x$ say, $\mathbb{Z_{p}}[x]$. Form its field of fractions $\mathbb{Z_{p}}(x)$ and then form the polynomial ring in the indeterminate $t$ say, $\mathbb{Z_{p}}(x)[t]$. Form its field of fractions $\mathbb{Z_{p}}(x)(t)$ and so on. This construction is used in several areas of maths including Galois theory to show the existence of inseparable polynomials.