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} |
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\section{$BM-8BBN(B} |
\section{$BM-8BBN(B} |
Line 194 $$\GCD(\prod_k g_k,\sum_i n_i g'_i \prod_{k\neq i}g_k) |
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Line 195 $$\GCD(\prod_k g_k,\sum_i n_i g'_i \prod_{k\neq i}g_k) |
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\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 |
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Line 411 $\GCD(\Tr(e),f) = 1 \Leftrightarrow$ $B$9$Y$F$N(B $i |
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\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. |
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\begin{al}(Cantor-Zassenhaus{\rm\cite{CZ}}) |
\begin{al}(Cantor-Zassenhaus{\rm\cite{CZ}}) |
Line 422 $Q \leftarrow \pi$ $B$N9TNsI=8=(B\\ |
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Line 423 $Q \leftarrow \pi$ $B$N9TNsI=8=(B\\ |
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$\{e_1 = 1, e_2, \cdots, e_r\} \leftarrow \Ker(Q-I)$ $B$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 $F \leftarrow F \backslash \{g\}$\\ |
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\> $(c_1,\cdots,c_r) \leftarrow$ $BMp?t%Y%/%H%k(B ($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 $F \leftarrow F \backslash \{g\}$\\ |
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\> \> $h \leftarrow \GCD(g,E)$\\ |
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\> \> if \= $h \neq 1,g$\\ |
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\> \> \> $F_1 \leftarrow F_1 \cup \{h,g/h\}$\\ |
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\> \> else \\ |
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\> \> \> $F_1 \leftarrow F_1 \cup \{g\}$\\ |
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\> \}\\ |
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\> $F \leftarrow F_1$ \\ |
\}\\ |
\}\\ |
return F |
return F |
\end{tabbing} |
\end{tabbing} |
Line 492 $q=p^n$ $B$G(B $p$ $B$,4qAG?t$H$9$k(B. $f=f_1f_2$ |
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Line 499 $q=p^n$ $B$G(B $p$ $B$,4qAG?t$H$9$k(B. $f=f_1f_2$ |
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$B$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, $BB?9`<0$ |
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Line 519 GF(q^d)\bigoplus GF(q^d)$ $B$N85$KBP$7(B, $BB?9`<0$ |
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$B$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 |
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\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}$ |
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Line 560 Input : $f(x) \in GF(q)[x]$, $q=p^n$, $f$ $B$OL5J?J}$ |
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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 $F \leftarrow F \backslash \{h\}$\\ |
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\> $g \leftarrow 2d-1$ $B<!$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 $F \leftarrow F \backslash \{h\}$\\ |
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\> \> $z \leftarrow \GCD(h,G)$\\ |
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\> \> if \= $z \neq 1,h$\\ |
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\> \> \> $F_1 \leftarrow F_1 \cup \{z,h/z\}$\\ |
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\> \> else \\ |
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\> \> \> $F_1 \leftarrow F_1 \cup \{h\}$\\ |
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\> \}\\ |
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\> $F \leftarrow F_1$ \\ |
\}\\ |
\}\\ |
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 |
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Line 1229 F(x) &=& x^{16} (x^2-28)^8 (x^2-20)^8 (x^2-8)^8 (x^2-1 |
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$B0lHL$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. |
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$B$^$?(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 $$\int f dx = \sum_i c_i\log r_i$$ |
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Line 1302 $$\int f dx = \sum_i c_i\log r_i$$ |
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$B$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. |
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\begin{pr}(Rothstein) |
\begin{pr}(Rothstein{\rm\cite{DAV}}) |
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$K$ $B$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 $$v_i=\GCD(n-c_id',d).$$ |
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Line 1318 $$v_i=\GCD(n-c_id',d).$$ |
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$v=\prod_{i=1}^n$ $B$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)$$ |
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Line 1340 $$g|(n-cd')=\sum_{j=1}^n (c_j-c)v_j'(v/v_j)$$ |
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$B$h$j(B, $g|(c_i-c)v_i'(v/v_i).$ $B$3$l$O(B $c_i=c$ $B$N$H$-$N$_2DG=(B. \qed |
$B$h$j(B, $g|(c_i-c)v_i'(v/v_i).$ $B$3$l$O(B $c_i=c$ $B$N$H$-$N$_2DG=(B. \qed |
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\begin{co} |
\begin{co} |
$K$ $B$rJ#AG?tBN(B $C$ $B$NItJ,BN$H$7(B, |
$K$ $B$rJ#AG?tBN(B $\C$ $B$NItJ,BN$H$7(B, |
$f(x)=n(x)/d(x)$, $n,d \in K[x]$, $\GCD(n,d)=1$, $\deg(n)<\deg(d)$ $B$G(B $d$ $B$OL5J?J}(B, $B%b%K%C%/$H$7(B, |
$f(x)=n(x)/d(x)$, $n,d \in K[x]$, $\GCD(n,d)=1$, $\deg(n)<\deg(d)$ $B$G(B $d$ $B$OL5J?J}(B, $B%b%K%C%/$H$7(B, |
$$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. |