groups in field

If  $(K,+,\cdot )$  is a field, then

• $(K,+)$  is the additive group of the field,

• $(K\setminus \{0\},\cdot )$  is the multiplicative group of the field.

Both of these groups are Abelian.

The former has always as a subgroup

$$\{n\cdot 1\mathrm{⋮}n\in \mathbb{Z}\},$$

the group of the multiples of unity.  This is, apparently, isomorphic to the additive group $\mathbb{Z}$ or ${\mathbb{Z}}_p$ 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  $({ℝ}_{+},\cdot )$  of the positive reals; the isomorphy is implemented e.g. by the isomorphism mapping  $x\mapsto 2^x$.

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   $f_i:K^{*}\to K^{*}$  defined by  $f_0(x):=x$,  $f_1(x):=-x$,  $f_2(x):=x^{-1}$,  $f_3(x):=-x^{-1}$.  The composition of functions is a binary operation of the set  $G=\{f_0,f_1,f_2,f_3\}$,  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