subfield criterion

Let $K$ be a skew field and $S$ its subset.  For $S$ to be a subfield of $K$, it’s necessary and sufficient that the following three conditions are fulfilled:

1. $S$ a non-zero element of $K$.

2. $a-b\in S$ always when  $a,b\in S$.

3. $a{b}^{-1}\in S$ always when  $a,b\in S$  and  $b\ne 0$.

Proof.  Because the conditions are fulfilled in every skew field, they are necessary.  For proving the sufficience, suppose now that the subset $S$ these conditions.  The condition 1 guarantees that $S$ is not empty and the condition 2 that  $(S,+)$  is an subgroup of  $(K,+)$;  thus all the required properties of addition for a skew field hold in $S$.  If $b$ is a non-zero element of $S$, then, according to the condition 3, we have  $0\ne 1=b{b}^{-1}\in S$.  Moreover,  $a\cdot 1=1\cdot a=a\in S$  for all  $a\in S\subseteq K$.  The laws of multiplication (associativity and left and distributivity over addition) hold in $S$ since they hold in whole $K$.  So $S$ fulfils all the postulates for a skew field.