Subfield/Direct/Definition

Subfield

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

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