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version 1.3, 2000/03/28 02:02:29

version 1.4, 2000/10/03 01:23:58

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%$OpenXM$

%$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}

\section{$BM-8BBN(B}

Line 411 $\GCD(\Tr(e),f) = 1 \Leftrightarrow$ $B$9$Y$F$N(B $i

\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.

$B$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.

\begin{al}(Cantor-Zassenhaus{\rm\cite{CZ}})

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

$B$V$H$-(B,

$\GCD(g^{(q^d-1)/2}-1,f)=f_1$ $B$^$?$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}

\proof

$f_1$, $f_2$ $B$,(B $d$ $B<!4{Ls$h$j(B,

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,

$\GCD(g^{(q^d-1)/2}-1,f)=f_1$ $B$^$?$O(B $f_2$ $B$H$J$k$N$O(B,

$$2(q^d-1)/2\cdot (q^d+1)/2 = (q^{2d}-1)/2$$

$B8D$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}

$q=2^n$ $B$H$9$k(B. $f=f_1f_2$ $B$G(B$f_1$, $f_2$ $B$,(B $f$ $B$N(B 2