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^{}
$$\{n\cdot 1\mathrm{\vdots}n\in \mathbb{Z}\},$$ 
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 $(\mathbb{R},+)$ of the reals is isomorphic to the multiplicative group $({\mathbb{R}}_{+},\cdot )$ 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 4group.
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  20130322 14:41:58 
Last modified on  20130322 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 