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鐃緒申鐃緒申鐃緒申鐃緒申鐃� I 鐃緒申鐃緒申 No.12

鐃緒申鐃緒申鐃緒申半鐃塾テ￥申鐃緒申:

\fbox{ネータ環の直積分解と加群}

鐃緒申鐃緒申 12.1   鐃縦器申鐃粛￥申鐃緒申鐃緒申 $R$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃塾とわ申, $R$ 鐃緒申直鐃緒申分鐃緒申

\begin{displaymath}R=R_1\times R_2 \times R_3 \times \dots \times R_n \end{displaymath}

鐃叔￥申 鐃緒申直鐃渋逸申鐃緒申 $R_i$ 鐃塾駈申鐃緒申鐃緒申鐃緒申 $1$ 鐃緒申 $0$ 鐃塾みでわ申鐃緒申茲�鐃淑わ申里鐃渋醐申澆鐃緒申襦�

PID 鐃薯イデワ申鐃緒申燃鐃獣わ申鐃処う鐃淑環につわ申鐃銃はわ申鐃叔にわ申辰半椶鐃緒申鐃緒申融劼鐃緒申里辰討鐃緒申襦�

鐃緒申鐃緒申 12.2   任鐃春の可器申鐃緒申 $R$ 鐃緒申鐃出わ申鐃緒申,

\begin{displaymath}\sqrt 0_R=\{x\in R ; x^n=0 \quad (\exists n \in {\mbox{${\Bbb Z}$}}_{>0})\} \end{displaymath}

鐃緒申 $R$ 鐃塾ワ申鐃叔ワ申鐃緒申任鐃緒申襦� 鐃盾し $R/\sqrt{0_R}$ 鐃緒申鐃緒申鐃緒申覆鐃出￥申$R$ 鐃塾駈申鐃緒申鐃緒申鐃緒申 $0$ 鐃緒申 $1$ 鐃塾みでわ申鐃暑。

鐃緒申鐃緒申 12.3   $($鐃緒申鐃緒申$)$$S$ 鐃緒申 PID 鐃叔￥申 $e\in S\setminus \{0\}$ 鐃淑ら、

\begin{displaymath}S/(e)\cong R_1\times R_2 \times \dots \times R_n \end{displaymath}

鐃淑わ申直鐃緒申分鐃緒申如鐃� $R_i/\sqrt{0_{R_i}}$ 鐃緒申鐃塾でわ申鐃緒申茲�鐃淑わ申里鐃渋醐申澆鐃緒申襦�

鐃緒申鐃緒申 12.4   鐃縦器申鐃緒申 $R$ 鐃緒申 $R=R_1\times R_2$ 鐃緒申直鐃緒申分鐃薯さわ申討鐃緒申鐃緒申箸鐃緒申鐃緒申, 任鐃春わ申 $R$-鐃獣件申 $M$ 鐃緒申, $R$-鐃獣件申鐃夙わ申鐃銃わ申直鐃緒申分鐃緒申

\begin{displaymath}M=M_1\oplus M_2 \end{displaymath}

鐃叔￥申$R$ 鐃緒申 $M$ 鐃舜の削申鐃術わ申鐃緒申鐃緒申直鐃緒申分鐃緒申砲鐃獣わ申鐃出角の件申

\begin{displaymath}(r_1,r_2). (m_1,m_2) =(r_1.m_1,r_2.m_2) \end{displaymath}

鐃祝書けわ申茲�鐃淑わ申里鐃緒申帖鐃�

PID $S$ 鐃緒申鐃出わ申鐃緒申, $S$-鐃獣件申鐃緒申鐃宿のよう鐃淑刻申造鐃緒申鐃縦わ申鐃緒申鐃緒申 単鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申辰鐃緒申鐃�

鐃緒申半鐃叔わ申鐃叔逸申鐃緒申分鐃緒申慮鐃塾�鐃緒申鐃祝つわ申鐃銃考わ申鐃緒申鐃緒申鐃緒申 $n$ 鐃緒申鐃叔逸申鐃緒申分鐃薯すわ申鐃緒申峺鐃緒申鐃重�鐃緒申鐃緒申法鐃熟￥申$2$ 鐃緒申鐃緒申 $\sqrt{n}$ 鐃殉でわ申 鐃緒申鐃緒申鐃叔順次鐃緒申辰討鐃緒申鐃緒申鐃祝￥申任鐃緒申襦� 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任鐃�$\sqrt{n}$ 鐃緒申鐃緒申鐃純す鐃緒申鐃緒申屬鐃緒申鐃緒申鐃緒申襦�

(鐃緒申法1) 鐃叔緒申鐃�$m$ 鐃縦わ申鐃緒申鐃緒申鐃粛�鐃渋個の随申鐃薯繰わ申鐃瞬わ申鐃緒申鐃緒申

\begin{displaymath}b_1,b_2,\dots,b_m,a_1,a_2\dots,a_n ,a_1,a_2,\dots,a_n,a_1,a_2,\dots,a_n,a_1,a_2,\dots \end{displaymath}

鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃塾とわ申,

\begin{displaymath}b_k=a_{2k}-a_k \end{displaymath}

鐃夙わ申鐃緒申鐃緒申, $b_l=b_{2l}=0$ 鐃夙なわ申 $l$ 鐃緒申存鐃淳わ申鐃暑。

鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申多鐃準式 $f$ 鐃緒申鐃緒申董鐃�

\begin{displaymath}x_{i+1}=f(x_i) \text{(mod $p$)} \end{displaymath}

鐃淑わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃祝わ申鐃塾種申法鐃緒申適鐃術わ申鐃緒申鐃�, $x=y \ (p)$ 鐃緒申鐃緒申 $x\neq y \ (n)$ 鐃淑わ申 $x,y$ 鐃緒申鐃緒申箸泙辰鐃�, $n$ 鐃緒申鐃叔逸申鐃述わ申鐃緒申泙鐃緒申腓�鐃緒申鐃暑。鐃緒申鐃緒申鐃� Pollard 鐃緒申 $\rho$ 法鐃夙わ申鐃緒申鐃緒申

鐃緒申鐃緒申 12.1   $\rho$ 法鐃殉わ申鐃熟わ申鐃緒申他鐃緒申鐃緒申法鐃緒申鐃術わ申鐃緒申 $n=7278202201270378983212653$ 鐃緒申鐃叔逸申鐃緒申鐃緒申鐃緒申茵� 鐃緒申鐃術わ申鐃緒申鐃緒申法鐃緒申, 鐃竣ワ申鐃緒申鐃緒申燹�鐃緒申鐃緒申喙孫垰鐃緒申屬鬚修鐃緒申襪鰹申函鐃�

(鐃緒申鐃緒申) $\rho$ 法鐃緒申 UBASIC 鐃叔書くと種申鐃塾よう鐃祝なる。 (鐃緒申鐃緒申鐃緒申鐃緒申鐃竣ワ申鐃緒申鐃緒申鐃緒申複鐃緒申鐃祝わ申鐃緒申里鐃緒申鬚韻襪随申鐃緒申 鐃純干鐃緒申率鐃塾逸申鐃緒申鐃竣ワ申鐃緒申鐃緒申鐃祝わ申鐃銃わ申鐃暑。 鐃緒申味鐃塾わ申鐃緒申佑浪鐃緒申匹鐃緒申襪�鐃緒申鐃緒申分鐃叔一か鐃緒申颪い討澆襪鰹申函鐃�) 

10  n=607143768775207   '分鐃薯すわ申鐃�
30  a=1:b=1        ' a=x_i, b=x_{2i} 鐃塾つわ申鐃叔わ申鐃暑。
50  while 1        ' 鐃緒申鐃緒申鐃緒申鐃緒申 wend (150 鐃緒申)鐃殉わ申無鐃渋ルー鐃緒申
60    a=fnf(a)@n
70    b=fnf(fnf(b))@n
80    x=gcd(a-b,n)
90    if x<>1 then print x,n  :end
150 wend 
160 end
1000 def fnf(k)   ' fnf() 鐃舜随申鐃緒申鐃緒申鐃�; fn... 鐃熟ユー鐃緒申鐃緒申鐃緒申愎鐃� 
1010 return(k*k+1)   ' 鐃緒申鐃緒申鐃緒申鐃循わ申鐃塾関随申鐃叔わ申茲わ申鐃�

ubasic 鐃淑どのプワ申鐃緒申鐃緒申鐃緒申鐃緒申鐃夙っわ申鐃緒申鐃夙わ申鐃淑わ申鐃述とは￥申 鐃瞬居申鐃塾ホ￥申鐃緒申據鐃緒申鐃緒申鐃緒申鐃�ubasic 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃宿わ申, ubasic 鐃緒申鐃緒申爐件申鐃緒申鐃緒申箸鐃緒申離侫鐃緒申鐃緒申鐃祝ワ申鐃縦�鐃淑どで上記鐃竣ワ申鐃緒申鐃緒申鐃緒申 鐃書いて￥申"rho.txt" 鐃淑どわ申鐃緒申存鐃緒申, 鐃緒申鐃塾わ申鐃緒申ubasic 鐃緒申動鐃緒申鐃緒申, load "rho.txt" 鐃夙わ申鐃緒申个茲わ申鐃緒申鐃緒申鐃緒申鐃� 鐃瞬居申鐃塾ペ￥申鐃緒申鐃祝は随申鐃緒申鐃緒申鐃緒申鐃緒申鐃春ワ申 mupad 鐃盾お鐃緒申鐃銃わ申鐃緒申里鐃�, 鐃緒申鐃緒申鐃夙ってわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,mupad 鐃緒申 ifactor 鐃舜随申鐃緒申鐃術わ申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申鐃祝器申単鐃緒申鐃緒申鐃緒申里如鐃�ifactor 鐃緒申鐃緒申鐃熟削申鐃緒申篭悗鐃緒申鐃夙わ申鐃暑。


Yoshifumi Tsuchimoto
2001-01-11