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version 1.1, 2000/03/01 02:25:51 version 1.6, 2001/02/27 08:07:24
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   %$OpenXM: OpenXM/doc/compalg/factor.tex,v 1.5 2001/02/07 07:17:46 noro Exp $
 \chapter{$BB?9`<0$N0x?tJ,2r(B}  \chapter{$BB?9`<0$N0x?tJ,2r(B}
   
 \section{$BM-8BBN(B}  \section{$BM-8BBN(B}
Line 194  $$\GCD(\prod_k g_k,\sum_i n_i g'_i \prod_{k\neq i}g_k)
Line 195  $$\GCD(\prod_k g_k,\sum_i n_i g'_i \prod_{k\neq i}g_k)
 \begin{al}($BL5J?J}J,2r(B)  \begin{al}($BL5J?J}J,2r(B)
 \label{sqfrmod}  \label{sqfrmod}
 \begin{tabbing}  \begin{tabbing}
 Input : $f(x) \in K[x]$ ($K$ $B$OI8?t(B $p>0$ $B$NBN(B)\\  Input : $f(x) \in K[x]$ ($K$ $B$OI8?t(B $p>0$ $B$NM-8BBN(B)\\
 Output : $f$ $B$NL5J?J}J,2r(B $f= g_1^{n_1}g_2^{n_2}\cdots$\\  Output : $f$ $B$NL5J?J}J,2r(B $f= g_1^{n_1}g_2^{n_2}\cdots$\\
 $F \leftarrow 1$\\  $F \leftarrow 1$\\
 if \= $f'\neq 0$ \{\\  if \= $f'\neq 0$ \{\\
Line 410  $\GCD(\Tr(e),f) = 1 \Leftrightarrow$ $B$9$Y$F$N(B $i
Line 411  $\GCD(\Tr(e),f) = 1 \Leftrightarrow$ $B$9$Y$F$N(B $i
 \end{center}  \end{center}
 $s \in GF(q)$ $B$H$9$k;~(B, $\Tr(s) \equiv 0, 1$ $B$J$k85$O$=$l$>$l(B $q/2$ $B8D(B.  $s \in GF(q)$ $B$H$9$k;~(B, $\Tr(s) \equiv 0, 1$ $B$J$k85$O$=$l$>$l(B $q/2$ $B8D(B.
 $B$h$C$F(B, $\GCD(\Tr(e),f) = f$, $\GCD(\Tr(e),f) = 1$ $B$J$k3NN((B  $B$h$C$F(B, $\GCD(\Tr(e),f) = f$, $\GCD(\Tr(e),f) = 1$ $B$J$k3NN((B
 $B$O$=$l$>$l(B $1/2^k$ $B8D$H$J$k(B. \qed\\  $B$O$=$l$>$l(B $1/2^k$ $B$H$J$k(B. \qed\\
 $B$3$l$i$r$b$H$K(B, $B<!$N$h$&$J%"%k%4%j%:%`$rF@$k(B.  $B$3$l$i$r$b$H$K(B, $B<!$N$h$&$J%"%k%4%j%:%`$rF@$k(B.
   
 \begin{al}(Cantor-Zassenhaus{\rm\cite{CZ}})  \begin{al}(Cantor-Zassenhaus{\rm\cite{CZ}})
Line 422  $Q \leftarrow \pi$ $B$N9TNsI=8=(B\\
Line 423  $Q \leftarrow \pi$ $B$N9TNsI=8=(B\\
 $\{e_1 = 1, e_2, \cdots, e_r\} \leftarrow \Ker(Q-I)$ $B$N(B $K$-$B4pDl(B\\  $\{e_1 = 1, e_2, \cdots, e_r\} \leftarrow \Ker(Q-I)$ $B$N(B $K$-$B4pDl(B\\
 if ($r = 1$) then return $F$\\  if ($r = 1$) then return $F$\\
 while\= ($|F| < r$) do \{\\  while\= ($|F| < r$) do \{\\
 \> $g \leftarrow F$ $B$N85(B, \quad $F \leftarrow F \backslash \{g\}$\\  
 \> $(c_1,\cdots,c_r) \leftarrow$ $BMp?t%Y%/%H%k(B ($c_i \in GF(q)$)\\  \> $(c_1,\cdots,c_r) \leftarrow$ $BMp?t%Y%/%H%k(B ($c_i \in GF(q)$)\\
 \> $e \leftarrow \sum c_ie_i$ \\  \> $e \leftarrow \sum c_ie_i$ \\
 \> if \= $p=2$\\  \> if \= $p=2$\\
 \>\> $E \leftarrow \Tr(e)$\\  \>\> $E \leftarrow \Tr(e) \bmod f$\\
 \>else\\  \>else\\
 \>\> $E \leftarrow e^{(q-1)/2}-1$\\  \>\> $E \leftarrow e^{(q-1)/2}-1 \bmod f$\\
 \> $h \leftarrow \GCD(g,E)$\\  \> $F_1 \leftarrow \emptyset$\\
 \> if $h \neq 1,g$\\  \> while \= ($F \neq \emptyset$) do \{\\
 \>\> $F \leftarrow F \cup \{h,g/h\}$\\  \>        \> $g \leftarrow F$ $B$N85(B, \quad $F \leftarrow F \backslash \{g\}$\\
   \>        \> $h \leftarrow \GCD(g,E)$\\
   \>        \> if \= $h \neq 1,g$\\
   \>        \> \> $F_1 \leftarrow F_1 \cup \{h,g/h\}$\\
   \>        \> else \\
   \>        \> \> $F_1 \leftarrow F_1 \cup \{g\}$\\
   \> \}\\
   \>   $F \leftarrow F_1$ \\
 \}\\  \}\\
 return F  return F
 \end{tabbing}  \end{tabbing}
Line 492  $q=p^n$ $B$G(B $p$ $B$,4qAG?t$H$9$k(B. $f=f_1f_2$ 
Line 499  $q=p^n$ $B$G(B $p$ $B$,4qAG?t$H$9$k(B. $f=f_1f_2$ 
 $B$D$N(B $d$ $B<!4{Ls0x;R$H$9$k(B. $GF(q)$ $B>e$N(B $B9b!9(B $2d-1$ $B<!<0(B $g$ $B$r%i%s%@%`$KA*(B  $B$D$N(B $d$ $B<!4{Ls0x;R$H$9$k(B. $GF(q)$ $B>e$N(B $B9b!9(B $2d-1$ $B<!<0(B $g$ $B$r%i%s%@%`$KA*(B
 $B$V$H$-(B,  $B$V$H$-(B,
 $\GCD(g^{(q^d-1)/2}-1,f)=f_1$ $B$^$?$O(B $f_2$  $\GCD(g^{(q^d-1)/2}-1,f)=f_1$ $B$^$?$O(B $f_2$
 $B$H$J$k3NN($O(B $1-1/(2q^d)$.  $B$H$J$k3NN($O(B $1/2-1/(2q^d)$.
 \end{pr}  \end{pr}
 \proof  \proof
 $f_1$, $f_2$ $B$,(B $d$ $B<!4{Ls$h$j(B,  $f_1$, $f_2$ $B$,(B $d$ $B<!4{Ls$h$j(B,
Line 512  GF(q^d)\bigoplus GF(q^d)$ $B$N85$KBP$7(B, $BB?9`<0$
Line 519  GF(q^d)\bigoplus GF(q^d)$ $B$N85$KBP$7(B, $BB?9`<0$
 $B$h$C$F(B, $2d-1$ $B0J2<$NB?9`<0(B $q^{2d}$ $B8D$N$&$A(B,  $B$h$C$F(B, $2d-1$ $B0J2<$NB?9`<0(B $q^{2d}$ $B8D$N$&$A(B,
 $\GCD(g^{(q^d-1)/2}-1,f)=f_1$ $B$^$?$O(B $f_2$ $B$H$J$k$N$O(B,  $\GCD(g^{(q^d-1)/2}-1,f)=f_1$ $B$^$?$O(B $f_2$ $B$H$J$k$N$O(B,
 $$2(q^d-1)/2\cdot (q^d+1)/2 = (q^{2d}-1)/2$$  $$2(q^d-1)/2\cdot (q^d+1)/2 = (q^{2d}-1)/2$$
 $B8D$G$"$j(B, $B3NN($O(B $1-1/(2q^{2d}).$ \qed  $B8D$G$"$j(B, $B3NN($O(B $1/2-1/(2q^{2d}).$ \qed
   
 \begin{pr}  \begin{pr}
 $q=2^n$ $B$H$9$k(B. $f=f_1f_2$ $B$G(B$f_1$, $f_2$ $B$,(B $f$ $B$N(B 2  $q=2^n$ $B$H$9$k(B. $f=f_1f_2$ $B$G(B$f_1$, $f_2$ $B$,(B $f$ $B$N(B 2
Line 553  Input : $f(x) \in GF(q)[x]$, $q=p^n$, $f$ $B$OL5J?J}$
Line 560  Input : $f(x) \in GF(q)[x]$, $q=p^n$, $f$ $B$OL5J?J}$
 Output : $f(x) = \prod f_i$, $f$ $B$N(B $B4{Ls0x;RJ,2r(B\\  Output : $f(x) = \prod f_i$, $f$ $B$N(B $B4{Ls0x;RJ,2r(B\\
 $r \leftarrow \deg(f)/d,\quad F \leftarrow \{f\}$\\  $r \leftarrow \deg(f)/d,\quad F \leftarrow \{f\}$\\
 while\= ($|F| < r$) do \{\\  while\= ($|F| < r$) do \{\\
 \> $h \leftarrow F$ $B$N(B $\deg(h)>d$ $B$J$k85(B,\quad $F \leftarrow F \backslash \{h\}$\\  
 \> $g \leftarrow 2d-1$ $B<!$N%i%s%@%`$JB?9`<0(B\\  \> $g \leftarrow 2d-1$ $B<!$N%i%s%@%`$JB?9`<0(B\\
 \> if \= $p=2$\\  \> if \= $p=2$\\
 \>\>$G \leftarrow \sum_{j=0}^{rd-1}g^{2^i}$\\  \>\>$G \leftarrow \sum_{j=0}^{rd-1}g^{2^i} \bmod f$\\
 \> else\\  \> else\\
 \>\> $G \leftarrow g^{(q^d-1)/2}-1$\\  \>\> $G \leftarrow g^{(q^d-1)/2}-1 \bmod f$\\
 \> $z \leftarrow \GCD(h,G)$\\  \> $F_1 \leftarrow \emptyset$\\
 \> if $z \neq 1,h$ \\  \> while \= ($F \neq \emptyset$) do \{\\
 \>\> $F \leftarrow F \cup \{z,h/z\}$\\  \>        \> $h \leftarrow F$ $B$N(B $\deg(h)>d$ $B$J$k85(B,\quad $F \leftarrow F \backslash \{h\}$\\
   \>        \> $z \leftarrow \GCD(h,G)$\\
   \>        \> if \= $z \neq 1,h$\\
   \>        \> \> $F_1 \leftarrow F_1 \cup \{z,h/z\}$\\
   \>        \> else \\
   \>        \> \> $F_1 \leftarrow F_1 \cup \{h\}$\\
   \> \}\\
   \>   $F \leftarrow F_1$ \\
 \}\\  \}\\
 return $F$\\  return $F$\\
 \end{tabbing}  \end{tabbing}
Line 1216  F(x) &=& x^{16} (x^2-28)^8 (x^2-20)^8 (x^2-8)^8 (x^2-1
Line 1229  F(x) &=& x^{16} (x^2-28)^8 (x^2-20)^8 (x^2-8)^8 (x^2-1
 $B0lHL$K(B, $BM-M}4X?t(B $f(x)=n(x)/d(x)$ ($n,d \in \Q[x]$) $B$NITDj@QJ,$O(B,  $B0lHL$K(B, $BM-M}4X?t(B $f(x)=n(x)/d(x)$ ($n,d \in \Q[x]$) $B$NITDj@QJ,$O(B,
 $f$ $B$NItJ,J,?tJ,2r$K$h$j7W;;$G$-$k(B. $B$7$+$7(B, $B$=$N$?$a$KJ,Jl(B $d$ $B$r(B  $f$ $B$NItJ,J,?tJ,2r$K$h$j7W;;$G$-$k(B. $B$7$+$7(B, $B$=$N$?$a$KJ,Jl(B $d$ $B$r(B
 1 $B<!0x;R$N@Q$KJ,2r$9$k$K$O(B $d$ $B$N:G>.J,2rBN$r5a$a$k$3$H$,I,MW$H$J$k(B.  1 $B<!0x;R$N@Q$KJ,2r$9$k$K$O(B $d$ $B$N:G>.J,2rBN$r5a$a$k$3$H$,I,MW$H$J$k(B.
   
 $B$^$?(B, $BITDj@QJ,<+BN$K(B, $d$ $B$NJ,2r$K$h$j8=$l$?Be?tE*?t$,8=$l$k$H$O(B  $B$^$?(B, $BITDj@QJ,<+BN$K(B, $d$ $B$NJ,2r$K$h$j8=$l$?Be?tE*?t$,8=$l$k$H$O(B
 $B8B$i$J$$(B.  $B8B$i$J$$(B.
 \begin{ex}  \begin{ex}
Line 1290  $$\int f dx = \sum_i c_i\log r_i$$
Line 1302  $$\int f dx = \sum_i c_i\log r_i$$
 $B$H=q$1$k(B. $B0lHL$K(B $c_i \in \Q, r_i \in \Q[x]$ $B$H$O8B$i$:(B, $B2?$i$+$NBe?tE*?t$r4^$`(B  $B$H=q$1$k(B. $B0lHL$K(B $c_i \in \Q, r_i \in \Q[x]$ $B$H$O8B$i$:(B, $B2?$i$+$NBe?tE*?t$r4^$`(B
 $B2DG=@-$,$"$k$,(B, $B$3$NBe?t3HBg$r:G>.8B$K$9$k$h$&$JI=<($r5a$a$?$$(B.  $B2DG=@-$,$"$k$,(B, $B$3$NBe?t3HBg$r:G>.8B$K$9$k$h$&$JI=<($r5a$a$?$$(B.
   
 \begin{pr}(Rothstein)  \begin{pr}(Rothstein{\rm\cite{DAV}})
   
 $K$ $B$rJ#AG?tBN(B $\C$ $B$NItJ,BN$H$7(B,  $K$ $B$rJ#AG?tBN(B $\C$ $B$NItJ,BN$H$7(B,
 $f(x)=n(x)/d(x)$, $n,d \in K[x]$, $\GCD(n,d)=1$, $\deg(n)<\deg(d)$ $B$G(B $d$ $B$OL5J?J}(B, $BL5J?J}$H$9$k(B. $B$3$N$H$-(B,  $f(x)=n(x)/d(x)$, $n,d \in K[x]$, $\GCD(n,d)=1$, $\deg(n)<\deg(d)$ $B$G(B $d$ $B$OL5J?J}(B, $BL5J?J}$H$9$k(B. $B$3$N$H$-(B,
 $$n/d = \sum_{i=1}^n c_i v_i'/v_i$$  $$n/d = \sum_{i=1}^n c_i v_i'/v_i$$
 $B$?$@$7(B $c_i \in \C$ $B$OAj0[$j(B, $v_i \in \C[x]$, $v_i$ $B$O%b%K%C%/(B, $BL5J?J}$G8_$$$KAG(B,  $B$?$@$7(B $c_i \in \C$ $B$OAj0[$J$j(B, $v_i \in \C[x]$, $v_i$ $B$O%b%K%C%/(B, $BL5J?J}$G8_$$$KAG(B,
 $B$H=q$1$?$J$i$P(B, $c_i$ $B$O(B  $B$H=q$1$?$J$i$P(B, $c_i$ $B$O(B
 $$R(z)=\res_x(n-zd',d) \in K[z]$$  $$R(z)=\res_x(n-zd',d) \in K[z]$$
 $B$N:,$G(B,  $B$N:,$G(B,
Line 1306  $$v_i=\GCD(n-c_id',d).$$
Line 1318  $$v_i=\GCD(n-c_id',d).$$
   
 $v=\prod_{i=1}^n$ $B$H$*$/$H(B,  $v=\prod_{i=1}^n$ $B$H$*$/$H(B,
 $$nv = d\sum_{i=1}^nc_iv_i'(v/v_i).$$  $$nv = d\sum_{i=1}^nc_iv_i'(v/v_i).$$
 $\GCD(n,d)=1$ $B$h$j(B $d|v.$ $B0lJ}$G(B, $v_i$|$B1&JU$h$j(B, $B$b$7(B $v{\not |}$  $\GCD(n,d)=1$ $B$h$j(B $d|v.$ $B0lJ}$G(B, $v_i|$$B1&JU$h$j(B, $B$b$7(B $v_i{\not |}d$
 $B$J$i$P(B $v_i|c_iv_i'(v/v_i)$ $B$H$J$k$,(B, $B$3$l$O(B $v_i$ $B$K4X$9$k>r7o$h$jIT2DG=(B.  $B$J$i$P(B $v_i|c_iv_i'(v/v_i)$ $B$H$J$k$,(B, $B$3$l$O(B $v_i$ $B$K4X$9$k>r7o$h$jIT2DG=(B.
 $B$h$C$F(B $v_i|d.$ $B7k6I(B $v|d$ $B$H$J$j(B $v=d.$ \qed\\  $B$h$C$F(B $v_i|d.$ $B7k6I(B $v|d$ $B$H$J$j(B $v=d.$ \qed\\
 \underline{claim 2} $v_i=\GCD(n-c_id',d).$  \underline{claim 2} $v_i=\GCD(n-c_id',d).$
Line 1328  $$g|(n-cd')=\sum_{j=1}^n (c_j-c)v_j'(v/v_j)$$
Line 1340  $$g|(n-cd')=\sum_{j=1}^n (c_j-c)v_j'(v/v_j)$$
 $B$h$j(B, $g|(c_i-c)v_i'(v/v_i).$ $B$3$l$O(B $c_i=c$ $B$N$H$-$N$_2DG=(B. \qed  $B$h$j(B, $g|(c_i-c)v_i'(v/v_i).$ $B$3$l$O(B $c_i=c$ $B$N$H$-$N$_2DG=(B. \qed
   
 \begin{co}  \begin{co}
 $K$ $B$rJ#AG?tBN(B $C$ $B$NItJ,BN$H$7(B,  $K$ $B$rJ#AG?tBN(B $\C$ $B$NItJ,BN$H$7(B,
 $f(x)=n(x)/d(x)$, $n,d \in K[x]$, $\GCD(n,d)=1$, $\deg(n)<\deg(d)$ $B$G(B $d$ $B$OL5J?J}(B, $B%b%K%C%/$H$7(B,  $f(x)=n(x)/d(x)$, $n,d \in K[x]$, $\GCD(n,d)=1$, $\deg(n)<\deg(d)$ $B$G(B $d$ $B$OL5J?J}(B, $B%b%K%C%/$H$7(B,
 $$R(z)=\res_x(n-zd',d) \in K[z]$$  $$R(z)=\res_x(n-zd',d) \in K[z]$$
 $B$H$9$k(B.  $B$H$9$k(B.
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