groups in field
If (K,+,⋅) is a field, then
-
•
(K,+) is the additive group
of the field,
-
•
(K∖{0},⋅) is the multiplicative group of the field.
Both of these groups are Abelian.
The former has always as a subgroup
{n⋅1⋮n∈ℤ}, |
the group of the multiples of unity. This is, apparently, isomorphic
to
the additive group ℤ or ℤp depending on whether the characteristic (http://planetmath.org/Characteristic) of the field is 0 or a prime number
p.
The multiplicative group of any field has as its subgroup the set E consisting of all roots of unity 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 (ℝ,+) of the reals is isomorphic to the multiplicative group (ℝ+,⋅) of the positive reals; the isomorphy is implemented e.g. by the isomorphism mapping x↦2x.
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):=,
, , . 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 |
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 |