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.

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

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 $0\neq 1=bb^{-1}\in S$ . 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	2013-03-22 16:26:34
Last modified on	2013-03-22 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