## 305 -Fields (5)

February 27, 2009

At the end of last lecture we stated a theorem giving an easy characterization of subfields of a given field ${\mathbb F}.$ We begin by proving this result.

Theorem 18. Suppose ${\mathbb F}$ is a field and $S\subseteq{\mathbb F}.$ If $S$ satisfies the following 5 conditions, then $S$ s a subfield of ${\mathbb F}:$

1. $S$ is closed under addition.
2. $S$ is closed under multiplication.
3. $-a\in S$ whenever $a\in S.$
4. $a^{-1}\in S$ whenever $a\in S$ and $a\ne0.$
5. $S$ has at least two elements.