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| Title of object: |
groups in field |
| Canonical Name: |
GroupsInField |
| Type: |
Topic |
| Created on: |
2004-10-06 16:17:51 |
| Modified on: |
2006-08-07 09:52:18 |
| Classification: |
msc:12E99, msc:20A05, msc:20F99, msc:20K99 |
| Defines: |
additive group of the field, multiplicative group of the field, additive group, multiplicative group |
Revision comment (for changes between this and next version):
Preamble:
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Content:
If \,$(K,\,+,\,\cdot)$\, is a field, then
\begin{itemize}
\item $(K,\,+)$ \,is the {\em additive group of the field},
\item $(K\setminus\{0\},\,\cdot)$ \,is the {\em multiplicative group of the field}.
\end{itemize}
Both of these groups are Abelian.
The former has always as a subgroup
$$\{n\!\cdot\!1: \,\,\,n\in\mathbb{Z}\},$$
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 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 $f_i$ from $K^*$ to $K^*$:\, $f_0(x) := x$,\, $f_1(x) := -x$,\, $f_2(x) := x^{-1}$,\, $f_3(x) := -x^{-1}$.\, 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. |
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