Subfield

Definition

Let $L$ be a field. A subfield of $L$ is a subring $K \subseteq L$ such that $a^{-1} \in K$ for every nonzero $a \in K$.

Remarks

Equivalently, $K$ contains $0$ and $1$, and $K$ is closed under addition, negation, multiplication and the inverse of nonzero elements. A subfield is a field with the operations of $L$ restricted to $K$. The rationals $\mathbb{Q}$ are a subfield of $\mathbb{R}$, and $\mathbb{R}$ is a subfield of $\mathbb{C}$. The intersection of any family of subfields of $L$ is a subfield of $L$. The intersection of all of them is the prime subfield.

Explanations

L1 human

A subfield is a smaller field that sits inside a bigger one. You can add, subtract, multiply and divide its elements without leaving it. The integers are a subring of $\mathbb{Q}$, but not a subfield, because $1/2 \notin \mathbb{Z}$.