first part of modulare arithmetik finished
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\chapter{Modulare Arithmetik}
|
||||
\section{Exkurs: Division mit Rest}
|
||||
Für $a,b\in \mathbb{Z},b\ne 0$ gibt es eindeutig bestimmte Element $q,r\in\mathbb{Z},0\le r<|b|$:
|
||||
$$\begin{aligned}
|
||||
a =b\cdot q+r\\
|
||||
a /_\mathbb{Z} b :&= q \\
|
||||
a\mod b :&= r \\
|
||||
\end{aligned}$$
|
||||
|
||||
\section{Der Ring $\mathbb{Z}_n$}
|
||||
Ein Ring $\mathbb{Z}_n$ ist definiert durch: $$\mathbb{Z}_n := {0,1,...,n-1}$$
|
||||
|
||||
\subsection{Addition und Multiplikation}
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
a +_{\mathbb{Z}_n} b :&= (a+b) \mod n\\
|
||||
a \cdot_{\mathbb{Z}_n} b :&= (a\cdot b) \mod n\\
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
\subsection{Inverse bezüglich der Addition}
|
||||
jedes $a\in \mathbb{Z}$ hat ein Inverses:
|
||||
$$ -a :=
|
||||
\begin{cases}
|
||||
0 &\text{für }a=0 \\
|
||||
n-a &\text{sonst}
|
||||
\end{cases}$$
|
||||
|
||||
\subsection{Subtraktion}
|
||||
Eine Subtraktion entspricht einer Addition mit der Inverse:
|
||||
$$a-_{\mathbb{Z}_n}b := a+_{\mathbb{Z}_n}(-b) \mod n$$
|
||||
|
||||
\subsection{Teiler, Vielfache}
|
||||
$b\in \mathbb{Z}$ teilt $a\in \mathbb{Z}$ falls ein $q\in \mathbb{Z}$ existiert mit:
|
||||
$$ a = b\cdot q$$
|
||||
man schreibt auch $b|a$
|
||||
|
||||
\subsubsection{Teilerregeln}
|
||||
\begin{enumerate}
|
||||
\item $a|0$ $\forall a\in \mathbb{Z}$
|
||||
\item $a|b \Leftrightarrow a|(-b)$
|
||||
\item $a|b \text{ und } a|c \Rightarrow a|(b+c)$
|
||||
\end{enumerate}
|
||||
|
||||
\subsection{Kongruenz}
|
||||
$a,b\in \mathbb{Z}$ sind \textit{kongruent modulo n}, falls $n\in \mathbb{N}|(a-b)$.
|
||||
Man schreibt auch $a\equiv b$
|
||||
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