a non-zero element of .
always when .
always when and .
Proof. Because the conditions are fulfilled in every skew field, they are necessary. For proving the sufficience, suppose now that the subset these conditions. The condition 1 guarantees that is not empty and the condition 2 that is an subgroup of ; thus all the required properties of addition for a skew field hold in . If is a non-zero element of , then, according to the condition 3, we have . Moreover, for all . The laws of multiplication (associativity and left and distributivity over addition) hold in since they hold in whole . So fulfils all the postulates for a skew field.
|Date of creation||2013-03-22 16:26:34|
|Last modified on||2013-03-22 16:26:34|
|Last modified by||pahio (2872)|