groups in field

If  (K,+,)  is a field, then

Both of these groups are AbelianMathworldPlanetmath.

The former has always as a subgroupMathworldPlanetmathPlanetmath


the group of the multiplesMathworldPlanetmathPlanetmath of unity.  This is, apparently, isomorphicPlanetmathPlanetmathPlanetmath to the additive group or p depending on whether the characteristic ( of the field is 0 or a prime numberMathworldPlanetmath p.

The multiplicative group of any field has as its subgroup the set E consisting of all roots of unityMathworldPlanetmath in the field.  The group E has the subgroup  {1,-1}  which reduces to {1} if the of the field is two.

Example 1.  The additive group  (,+)  of the reals is isomorphic to the multiplicative group  (+,)  of the positive reals; the isomorphy is implemented e.g. by the isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath mapping  x2x.

Example 2.  Suppose that the of K is not 2 and denote the multiplicative group of K by K*.  We can consider the four functions   fi:K*K*  defined by  f0(x):=x,  f1(x):=-x,  f2(x):=x-1,  f3(x):=-x-1.  The composition of functions is a binary operationMathworldPlanetmath of the set  G={f0,f1,f2,f3},  and we see that G 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
Canonical name GroupsInField
Date of creation 2013-03-22 14:41:58
Last modified on 2013-03-22 14:41:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 24
Author pahio (2872)
Entry type Topic
Classification msc 20K99
Classification msc 20F99
Classification msc 20A05
Classification msc 12E99
Related topic Klein4Group
Related topic Klein4Ring
Related topic GroupsOfRealNumbers
Defines additive group of the field
Defines multiplicative group of the field
Defines additive group
Defines multiplicative group