# groups in field

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

• $(K,\,+)$  is the ,

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

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