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Homesubfield criterion

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# 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. $S$ contains 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 $S$ satisfies 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 right distributivity over addition) hold in $S$ since they hold in whole $K$. So $S$ fulfils all the postulates for a skew field.

## Mathematics Subject Classification

12E99*no label found*12E15

*no label found*

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