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Revision difference : groups in field |
| Version 8 |
Version 7 |
| If \,$(K,\,+,\,\cdot)$\, is a field, then |
If \,$(K,\,+,\,\cdot)$\, is a field, then |
| \begin{itemize} |
\begin{itemize} |
| \item $(K,\,+)$ \,is the {\em additive group of the field}, |
\item $(K,\,+)$ \,is the {\em additive group of the field}, |
| \item $(K\setminus\{0\},\,\cdot)$ \,is the {\em multiplicative group of the field}. |
\item $(K\setminus\{0\},\,\cdot)$ \,is the {\em multiplicative group of the field}. |
| \end{itemize} |
\end{itemize} |
| Both of these groups are abelian. |
Both of these groups are abelian. |
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| The former has always as a subgroup |
The former has always as a subgroup |
| $$\{n\cdot 1: \,\,\,n\in\mathbb{Z}\},$$ |
$$\{n\cdot 1: \,\,\,n\in\mathbb{Z}\},$$ |
| the group of the multiples of unity. \,This is, apparently, isomorphic to |
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 additive group $\mathbb{Z}$ or $\mathbb{Z}_p$ depending on whether the \PMlinkname{characteristic}{Characteristic} of the field is 0 or a prime number $p$. |
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| The multiplicative group of any field has $\{1, -1\}$ as a subgroup; this reduces to $\{1\}$ if the \PMlinkescapetext{characteristic} of the field is two. |
The multiplicative group of any field has $\{1, -1\}$ as a subgroup; this reduces to $\{1\}$ if the \PMlinkescapetext{characteristic} of the field is two. |
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| \textbf{Example.} \,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 the Klein's 4-group. |
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