groups in field GroupsInField 2004-10-06 16:17:51 2009-10-06 09:10:07 Topic field additive group of the field multiplicative group of the field additive group multiplicative group % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here If \,$(K,\,+,\,\cdot)$\, is a field, then \begin{itemize} \item $(K,\,+)$ \,is the {\em additive group of the field}, \item $(K\!\smallsetminus\!\{0\},\,\cdot)$ \,is the {\em multiplicative group of the field}. \end{itemize} 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 \PMlinkname{characteristic}{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 \PMlinkescapetext{characteristic} of the field is two.\, \textbf{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 \,$x\mapsto 2^x$. \textbf{Example 2.}\, Suppose that the \PMlinkescapetext{characteristic} 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. \textbf{Note.}\, One may also speak of the {\em additive group} of any {\em ring}.\, Every ring contains also its group of units.