# groups in field

If  $(K,\,+,\,\cdot)$  is a field, then

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

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

Both of these groups are Abelian.

The former has always as a subgroup

 $\{n\!\cdot\!1\vdots\,\,\,n\in\mathbb{Z}\},$

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 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