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version 1.1.1.1, 2000/03/01 02:25:51 version 1.4, 2000/10/03 01:23:58
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   %$OpenXM: OpenXM/doc/compalg/factor.tex,v 1.3 2000/03/28 02:02:29 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 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 492  $q=p^n$ $B$G(B $p$ $B$,4qAG?t$H$9$k(B. $f=f_1f_2$ 
Line 493  $q=p^n$ $B$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 513  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 1216  F(x) &=& x^{16} (x^2-28)^8 (x^2-20)^8 (x^2-8)^8 (x^2-1
Line 1217  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 1290  $$\int f dx = \sum_i c_i\log r_i$$
 $B$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 1306  $$v_i=\GCD(n-c_id',d).$$
   
 $v=\prod_{i=1}^n$ $B$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)$$
Line 1328  $$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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