Prime subfield

Definition

Let $K$ be a field. The prime subfield of $K$ is the intersection of all the subfields of $K$. It is the smallest subfield of $K$.

Remarks

If the characteristic of $K$ is a prime $p$, then the prime subfield is isomorphic to $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$. If the characteristic of $K$ is $0$, then the prime subfield is isomorphic to $\mathbb{Q}$. The prime subfield of $\mathbb{R}$ is $\mathbb{Q}$.

Explanations

L1 human

Every subfield contains $0$ and $1$, and so it contains all the sums $1 + 1 + \dots + 1$ and their quotients. These elements form the prime subfield. In the real numbers, they are exactly the rational numbers.