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单词 GroupsInField
释义

groups in field


If  (K,+,)  is a field, then

  • (K,+)  is the additive groupMathworldPlanetmath of the field,

  • (K{0},)  is the multiplicative group of the field.

Both of these groups are AbelianMathworldPlanetmath.

The former has always as a subgroupMathworldPlanetmathPlanetmath

{n1n},

the group of the multiplesMathworldPlanetmathPlanetmath of unity.  This is, apparently, isomorphicPlanetmathPlanetmathPlanetmath tothe additive group or p depending on whether the characteristic (http://planetmath.org/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.

Titlegroups in field
Canonical nameGroupsInField
Date of creation2013-03-22 14:41:58
Last modified on2013-03-22 14:41:58
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id24
Authorpahio (2872)
Entry typeTopic
Classificationmsc 20K99
Classificationmsc 20F99
Classificationmsc 20A05
Classificationmsc 12E99
Related topicKlein4Group
Related topicKlein4Ring
Related topicGroupsOfRealNumbers
Definesadditive group of the field
Definesmultiplicative group of the field
Definesadditive group
Definesmultiplicative group
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