Subfield/Direct/Definition

A subset ${\displaystyle {}K\subseteq L}$ of a field ${\displaystyle {}L}$ is called a subfield of ${\displaystyle {}L}$, if the following properties hold.

1. We have ${\displaystyle {}0,1\in K}$.
2. For ${\displaystyle {}a,b\in K}$ also ${\displaystyle {}a+b\in K}$ holds.
3. For ${\displaystyle {}a,b\in K}$, also ${\displaystyle {}a\cdot b\in K}$ holds.
4. For ${\displaystyle {}a\in K}$, we also have ${\displaystyle {}-a\in K}$.
5. For ${\displaystyle {}a\in K}$, ${\displaystyle {}a\neq 0}$, we also have ${\displaystyle {}a^{-1}\in K}$.