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Dann existiert - ein $k \in \Z$ mit $ax = b + kn$, also mit $b = ax - kn$. Wegen + (i) $\implies$ (ii): Sei $x \in \Z$ mit $ax \equiv b$ $(\text{mod }n)$. Dann gilt + $n \mid (ax - b) \implies \exists k \in \Z$ mit $ax - b = kn$, also mit $b = ax - kn$. Wegen $\text{ggT}(a,n) \mid a$ und $\text{ggT}(a,n) \mid n$ folgt $\text{ggT}(a,n) \mid b$. (ii) $\implies$ (i): Es gelte $\text{ggT}(a,n) \mid b$. Aus dem erweiterten Euklidischen Algorithmus @@ -172,7 +172,8 @@ im Ring $\Z / n \Z$ zu charakterisieren.} $\overline{a} \cdot \overline{x} = \overline{1}$ in $\Z / n \Z$, welche genau dann lösbar ist, wenn $\overline{x}$ eine Einheit in $\Z / n \Z$ ist. - (i) $\iff$ (iii): Folgt mit $b = 1$ aus \ref{lemma:kongruenz}. + (i) $\iff$ (iii): $\text{ggT}(a,n) = 1 \iff \text{ggT}(a,n) \mid 1$. Folgt mit $b = 1$ aus + \ref{lemma:kongruenz}. \end{proof} \begin{bsp} @@ -217,6 +218,22 @@ $0$ und $n$.} \end{enumerate} \end{bsp} +\begin{lemma} + Es sei $p$ eine Primzahl und $n \in \N$. Dann gilt + \[ + \varphi(p^{n}) = p^{n-1}(p-1) + .\] +\end{lemma} + +\begin{proof} + $\exists $ genau $p^{n-1}$ Zahlen $a$ mit $0 \le a < p^{n}$ und $\text{ggT}(a,n) > 1$, denn: + Primfaktorzerlegung von $p^{n} = \underbrace{p \cdot p \cdot \ldots \cdot p}_{n-\text{mal}}$ + $\implies$ Zahlen $a$ sind die $p^{n-1}$ Vielfachen von $p$, also + $0 \cdot p, 1\cdot p, \ldots, (p^{n-1} - 1)\cdot p \implies \varphi(p^{n})=p^{n} - p^{n-1} = p^{n-1}(p-1)$. +\end{proof} + +Notation für Gruppen: Verknüpfung multiplikativ und neutrales Element ist $1$. + \begin{lemma} \label{lemma:endlichegruppe} Es sei $G$ eine endliche abelsche Gruppe und $g \in G$. Dann gilt @@ -273,4 +290,5 @@ $0$ und $n$.} Falls $\overline{a} = \overline{0}$ ist $\overline{a}^{p} = \overline{0}^{p} = \overline{0} = \overline{a}$. \end{enumerate} \end{proof} + \end{document} diff --git a/sose2020/theo/uebungen/theo2.pdf b/sose2020/theo/uebungen/theo2.pdf index 707a5b2a47b58cdd49a60bd0b53287636e04c59e..f12a47aa62b85827fd738e44e04fd63c97a9935e 100644 GIT binary patch delta 3286 zcmV;{3@P*9#RlQV2CzU612;A>lc6Rkf8|?UZ(O$#eb=wJk1MqIn%^7-DA2T)iv$Re zra~TU2gS--uQ!q8W+f#>AKbs5L*9=i@AC5QR%;tbUo3kyJ0xe$%$yn0&o00FmRIZh zGghrDR{iblFPCS}UT|Hx#u;I%%eR#hjTRMg*>G#C%j@dZ;_}_<{dp~=Tzqw2e+ycC z^+o%y*HWsh$Gi3Y)y?wZq5Vui!bqnVe|@*z+&^rVcm8v=_-3_zSlylfaQXGw#pT)0 zXB^A3iucPg*GMO-H@9c6eqhx#e)V<5@DWwN2Ep4(Nr+hd^``pA+24l)OdmwEMp#~1 zs~aIzdy>~ovp4vIhmG{#tzOldf3d~$&8j`a%U{oN$|Vl=(|OGo+vVHta5zx=m-TAB zx(c>f-L;=wKW^XnZ@u%Hu@fHcgkvkrimH~pk-FQ;2QOIF_(WDUjKHb={AH_v;_u@m z!Yf6mxQHqN&0?}Y9*ce!0ioL263!R5ZL+5j=e{L79VUF*6DRs5BZtQA?4LU0+C^5jfsNfROZR>VF@;NgN zFeJ8^#EGSF-cye=}x6&=@%=`gg|Y zup_|I$IJVhb(cD>H`~=uIm;WrO$8sxMsPWiB;1inVn6<-gG(gAe>IL>DY_3LhmJ9B zeegC$xPsw$SiX+LUNf1A7{J{Q`DE~B;&l=Pa>8Yr&KVxn9L+X`ntgpK?2}X zC6gn#ZNcK;7z{W!+8_&3V(~OZAW$J<^of8@hx&z%^h+&-So|Ru(b)b_aOUwcl&mBk zB=VAja}3vW0T>CD${XsGQ@Qdc((*P=POu;>&0H5HUnKnQf7uI>z)v!bm@jy6bnxSNiJlQXxV8?@;uTu;1U^~EvE?P#XS@oTHKuT#Rd$r{nu*aDWtKB zZ#{o}?^!Z_K6vK2f#C9{6}!!sYl(`d>>kJS@Kc5oR`(*~WMnj<$3;V>5{xo=al{Rc zk{^R-exD;Be@y7u+!-Ab@j1l7hjT@?FVU2wO%isT#YWz>jh4e|zXS(uo7N%Em)<48 zbL-8`L;K;y-Dt?cT)^^cG^DFQ6XmMu@%Kae8?kqM!| zNl#NMgP=u)HjD=zTu6kV>uQt<6JQvmRHZBJkiTS{e+ePv1x-%`{+Z8xWCLR%K^ZT^ zNu>1x>N8i9`4%BYZ4vKKpiQiU`NAH9;OY61_Vb}@QCMMF}ohGh%&?YA)d=5oDse0RK>o_b{P&(0k@84 zI@w4jt448`RyNjfC1)yI+wqihiupLzXrdz3f7~Ia(9DE{(HDc~!}uvo+b+*ZEw-z> zKqcB5gio(m+qMV&{v7mXx!o+UZhWcgMs!i}Neqs@j`{oqZp_SaF_q$6nj+Nzi?__X z)^?u#F>`znL^GIaBH^&SwUfmXbN`shNseJ3D1e-xu!w9!K{ADQGE(VP8FcpTia%#! 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