The prime subfield of a field $F$ is the intersection of all subfields of $F$ , or equivalently the smallest subfield of $F$ . It can also be constructed by taking the quotient field of the additive subgroup of $F$ generated by the multiplicative identity $1$ .

If $F$ has characteristic $p$ where $p > 0$ is a prime, then the prime subfield of $F$ is isomorphic to the field $\mathbb{Z}/p\mathbb{Z}$ of integers mod $p$ . When $F$ has characteristic zero, the prime subfield of $F$ is isomorphic to the field $\mathbb{Q}$ of rational numbers.