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 nonzero element of $K$.

2.
$ab\in S$ 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 nonzero 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.
Title  subfield criterion 

Canonical name  SubfieldCriterion 
Date of creation  20130322 16:26:34 
Last modified on  20130322 16:26:34 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  7 
Author  pahio (2872) 
Entry type  Theorem^{} 
Classification  msc 12E99 
Classification  msc 12E15 
Related topic  FieldOfAlgebraicNumbers 