groups in field
If is a field, then
Both of these groups are Abelian.
The former has always as a subgroup
the group of the multiples of unity. This is, apparently, isomorphic to the additive group or depending on whether the characteristic (http://planetmath.org/Characteristic) of the field is 0 or a prime number .
The multiplicative group of any field has as its subgroup the set consisting of all roots of unity in the field. The group has the subgroup which reduces to if the of the field is two.
Example 2. Suppose that the of is not 2 and denote the multiplicative group of by . We can consider the four functions defined by , , , . The composition of functions is a binary operation of the set , and we see that is isomorphic to Klein’s 4-group.
Note. One may also speak of the additive group of any ring. Every ring contains also its group of units.
|Title||groups in field|
|Date of creation||2013-03-22 14:41:58|
|Last modified on||2013-03-22 14:41:58|
|Last modified by||pahio (2872)|
|Defines||additive group of the field|
|Defines||multiplicative group of the field|