p“êõ‘ªõ‘ªVgõ‘ªVg½Ân∈N§K´˜‡ê�§x´˜‡/ªÎÒ£½¡™½¤/ªLˆªanxn+an−1xn−1+···+a1x+a0§Ù¥§a0,a1,···,an∈K§¡�ê�Kþ˜‡˜õ‘ª¡Ù¥aixi�1ig‘§¡ai�1ig‘Xêan6=0ž§¡dõ‘ª�˜‡ngõ‘ª§ÙgênP�n=deg anxn+an−1xn−1+···+a1x+a0Ï~§^f(x),g(x),···½f,g,···5L«õ‘ªKþõ‘ªNP�K[x]gõ‘ª´˜‡š~êõ‘ª§z‘Xêþ�õ‘ª£=0~êõ‘ª¤¡�õ‘ª§Ùgê½Â�−∞õ‘ªf(x)†g(x)¤kÓg‘þƒž§¡f(x)†g(x)ƒ§P�f(x)=g(x)p“êõ‘ªõ‘ªVgõ‘ª$Žµõ‘ª“Ú”µéAXêƒ\¶“”µéAXêƒ~¶“È”µ¦^©Æ¿Ü¿Óa‘äNXeµf(x)=n∑i=0aixi§g(x)=m∑j=0bjxj§f(x)±g(x)=max{n,m}∑i=0(ai±bi)xi¶f(x)g(x)=n+m∑k=0(∑i+j=kaibj)xkgêúªµdeg(f(x)±g(x))≤max(degf(x),degg(x))§degf(x)g(x)=degf(x)+degg(x)p“êõ‘ªØ½n(‘{Ø{)f(x),06=g(x)∈K[x]§K3�˜˜éq(x),r(x)∈K[x]§¦f(x)=q(x)g(x)+r(x)¤á§Ù¥degr(x)degg(x)‘{Ø{¥q(x)¡�g(x)Øf(x)û§r(x)¡�g(x)Øf(x){ªr(x)=0ž§·‚¡g(x)Øf(x)½Â(Ø)f(x),06=g(x)∈K[x]§e3q(x)∈K[x]§¦f(x)=g(x)q(x)K¡g(x)Øf(x)§P�“g(x)|f(x)”eþ¡q(x)Ø3§K¡g(x)ØØf(x)§P�“g(x)∤f(x)”g(x)|f(x)ž§¡g(x)�f(x)Ϫ§f(x)�g(x)ªíØf(x),06=g(x)∈K[x]§Kg(x)|f(x)…=g(x)Øf(x){ª�p“êõ‘ªØØke5Ÿµ·K(Ø5Ÿ)1šõ‘ªf(x)ØÙgC¶2D45µef(x)|g(x)§g(x)|h(x)§Kf(x)|h(x)¶3ef(x)|g(x)§g(x)|f(x)§Kf(x)=cg(x)§Ù¥06=c∈K¶4ef(x)|gi(x)§i=1,2,···,m§Kf(x)|[u1(x)g1(x)+u2(x)g2(x)+···+um(x)gm(x)]Ù¥ui(x)∈K[x]¶5ef(x)|g(x)§Kcf(x)|g(x)§Ù¥06=c∈K~1eg(x)|(f1(x)+2f2(x))…g(x)|(2f1(x)+f2(x))§Kg(x)|f1(x)§g(x)|f2(x)2¦a§b§¦x2+x+a|x3+bx+1p“êõ‘ª�ŒúϪ½Â(úϪ)XJõ‘ªd(x)Q´f(x)Ϫ§�´g(x)Ϫ§K¡d(x)´f(x)Úg(x)˜‡úϪf(x)!g(x)∈K[x]§Ø�§ed(x)∈K[x]§÷v1d(x)´f(x)Úg(x)úϪ¶2f(x)§g(x)úϪѴd(x)Ϫ§=ed1(x)|f(x)∧d1(x)|g(x)§Kd1(x)|d(x)§K¡d(x)´f(x)Úg(x)�ŒúϪd(x)Ä‘Xê�1£¡�Ä1¤§KPd(x)=(f(x),g(x))Ún(1)e06=f(x)|g(x)∈K[x]§Kf(x)´f(x)†g(x)˜‡�ŒúϪ(2)3‘{Ø{f(x)=q(x)g(x)+r(x)¥f(x)†g(x)úϪ8Úg(x)†r(x)úϪ8ƒÓp“êõ‘ª�ŒúϪÚnµ(1)e06=f(x)|g(x)∈K[x]§Kf(x)´f(x)†g(x)˜‡�ŒúϪ(2)3‘{Ø{f(x)=q(x)g(x)+r(x)¥f(x)†g(x)úϪ8Úg(x)†r(x)úϪ8ƒÓ½nf(x),g(x)∈K[x]Ø�§Kf(x)†g(x)�ŒúϪd(x)3§…3u(x),v(x)∈K[x]§¦d(x)=f(x)u(x)+g(x)v(x)y²µÎ=ƒØ{µ|^c¡Ún§f(x)Úg(x)¥k˜‡Ø,˜‡ž§Xg(x)|f(x)ž§Kg(x)��ŒúϪ§…g(x)=f(x)×0+g(x)×1dÚn£2¤§‡¦f(x)†g(x)�ŒúϪ§�I¦g(x)†r(x)�ŒúϪ§…džr(x)´f(x)†g(x)|ܶ2Î=§^g(x)†r(x)‘{Ø{{ª§XdE§(Ø5�ŒúϪ†ê�Ã'p“êõ‘ª�ŒúϪ½Â(pƒõ‘ª)f(x)§g(x)∈K[x]§e(f(x),g(x))=1§K¡f(x)§g(x)pƒ½nK[x]¥ü‡õ‘ªf(x)§g(x)pƒ¿‡^‡´3u(x),v(x)∈K[x]§¦u(x)f(x)+v(x)g(x)=1þã½n´k'ü‡õ‘ªpƒ�‡'Xª½n1XJ(f(x),g(x))=1§…f(x)|g(x)h(x)§Kf(x)|h(x)¶2XJf1(x)|g(x),f2(x)|g(x)§…(f1(x),f2(x))=1§Kf1(x)f2(x)|g(x)¶3XJ(f1(x),g(x))=1§(f2(x),g(x))=1§K(f1(x)f2(x),g(x))=1p“êõ‘ª�ŒúϪ½Âf1(x),f2(x),...,fn(x)∈K[x]Ø�§d(x)∈K[x]÷vµ1d(x)|fi(x)§∀1≤i≤n¶2XJd1(x)|fi(x)§∀1≤i≤n§Kd1(x)|d(x)§K¡d(x)�f1(x),f2(x),...,fn(x)�ŒúϪ§Ä˜�ŒúϪP�(f1(x),f2(x),...,fn(x))´§(f1(x),f2(x),...,fn(x))=((···((f1(x),f2(x)),f3(x)),...),fn(x))¿^8B{Œy3u1(x),u2(x),...,un(x)∈K[x]§¦(f1(x),f2(x),...,fn(x))=f1(x)u1(x)+f2(x)u2(x)+···+fn(x)un(x)p“êõ‘ª�ŒúϪ½n(¥I�{½n)f1(x),...,fn(x)∈K[x]üüpƒ§Ké?¿‰½a1(x),...,an(x)∈K[x]§73g(x),q1(x),...,qn(x)∈K[x]§¦g(x)=fi(x)qi(x)+ai(x),(P�g(x)≡ai(x)modfi(x)§i=1,...,ny²µfi(x)†f1(x)···fi−1(x)fi+1(x)···fn(x)pƒ§¤±3ui(x),vi(x)¦fi(x)ui(x)+f1(x)···fi−1(x)fi+1(x)···fn(x)vi(x)=1u´é?¿1≤k≤nn∑i=1ai(x)vi(x)∏j6=ifj(x)=n∑i6=kai(x)vi(x)∏j6=ifj(x)+ak(x)f1(x)···fk−1(x)fk+1=n∑i6=kai(x)vi(x)∏j6=ifj(x)+ak(x)−fk(x)uk(x)g(x)=∑ni=1ai(x)vi(x)∏j6=ifj(x)=ŒŒ§z‡ù)†‰½g(x)ƒ∏nj=1fj(x)ªp“êõ‘ª�ŒúϪ¥I�{½núc 5šfŽ²6¥k“ÔØꔯKµ“8kÔØÙê§nnêƒ{§ÊÊêƒ{n§ÔÔêƒ{§¯ÔAÛº”‰�“233”�Ò´¦Ó{ª|x≡2mod3x≡3mod5x≡2mod7¶²Š§Œ ^y‚‰ÑTK){µ“nÓ1Ô›D§Êärsö˜{§Ôf쌧ØzÊB”(3,5×7)=1§3×(−23)+35×2=1¶(5,21)=1§5×(−4)+21×1=1¶(7,15)=1§7×(−2)+15×1=1¶¤±§‡¦ê´2×35×2+3×21×1+2×15×1+3×5×7×k=233+105kp“êõ‘ªÏª©)½ÂgêŒu½u1õ‘ªp(x)∈K[x]§eØUL«¤K[x]¥ü‡gêþ'p(x)$õ‘ªƒÈ§K¡p(x)�ê�Kþ˜‡ØŒõ‘ª51˜gõ‘ªo´ØŒ¶2õ‘ªŒ5†Xê�ƒ'§Xx3−23QþØŒ§3RþŒ¶x2+13RþØŒ§3Cþ´Œ3p(x)´ê�KþØŒõ‘ª§f(x)∈K[x]§K(p(x),f(x))|p(x)§p(x)Ϫ�k~ê½p(x)ª§¤±(p(x),f(x))=1½(p(x),f(x))=cp(x)§¤±k½´p(x)†f(x)pƒ§½´p(x)|f(x)¶'X�köƒ˜p“êõ‘ªÏª©)Únp(x)´KþØŒõ‘ª§XJp(x)|f(x)g(x)§K7kp(x)|f(x)½p(x)|g(x)5XJØŒõ‘ªp(x)Øf1(x)f2(x)···fn(x)§´7ØÙ¥ƒ˜½n(Ϫ©)½n)ê�KþgêŒu½u1õ‘ªf(x)Ñ©)¤˜ØŒõ‘ª¦È¶…ù«©)ª3ØOgSž´�˜=µef(x)=cp1(x)p2(x)···pr(x)=cq1(x)q2(x)···qs(x),K7kr=s§…k˜‡ü(i1i2···ir)§¦pk(x)=qik(x)§k=1,2,...,rp“êõ‘ªÏª©)½n(õ‘ªÏª©)½n§½n¥©)ª¡�f(x)Oƒ©)ª)ê�Kþ?˜õ‘ªf(x)þŒ©)�f(x)=cpr11(x)pr22(x)···prnn(x),Ù¥c∈K§p1(x),p2(x),···,pn(x)∈K[x]´pÉĘ،õ‘ª§r1,r2,···,rn∈Z+¶e06=g(x)´˜‡Ïª§Kg(x)=c1ps11(x)ps22(x)···psnn(x),Ù¥c1∈K§s1,s2,...,sn∈Z§si≤ri(i=1,...,n)Šâ±þ½n§·‚˜„/k c1pr11(x)pr22(x)···prnn(x),c2ps11(x)ps22(x)···psnn(x)=pmin{r1,s1}1(x)pmin{r2,s2}2(x)···pmin{rn,sn}n(x)ef(x)†g(x)pƒ…=f(x)Úg(x)©)ª¥ØÑyÓØŒõ‘ªp“êõ‘ªÏª©)Ϫ½Â(Ϫ)p(x),f(x)∈K[x]§…p(x)ØŒ§ek∈Z+§pk(x)|f(x)…pk+1(x)∤f(x)§K¡p(x)´f(x)k-Ϫk≥2ž§¡p(x)´f(x)Ϫ،õ‘ªp(x)´õ‘ªf(x)k-Ϫ…=3õ‘ªg(x)§¦f(x)=pk(x)g(x)§…p(x)∤g(x)½nf(x)3ê�KþOƒ©)�f(x)=cpr11(x)pr22(x)···prnn(x)§Ù¥pi(x)Ę،!§ri∈Z+¶K f(x),f′(x)=pr1−11(x)pr2−12(x)···prn−1n(x)AO/§f(x)vkϪ…=(f(x),f′(x))=1§=f(x)†f′(x)pƒp“êõ‘ªÏª©)íØ1XJØŒõ‘ªp(x)´õ‘ªf(x)˜‡k-Ϫ£k≥1¤§Kp(x)´f(x)§f′(x),...,f(k−1)(x)Ϫ§Ø´f(k)(x)Ϫ2ØŒõ‘ªp(x)´õ‘ªf(x)Ϫ…=p(x)´f(x)†f′(x)úϪ~äõ‘ªf(x)=x4−x3−7x2+13x−6kvkϪíØf(x)z‡ØŒÏªÑy3f(x)(f(x),f′(x))¥TИgp“êõ‘ªõ‘ª¼êõ‘ª¼êf(x)=anxn+an−1xn−1+···+a1x+a0´ê�Kþõ‘ª§½Â¼êf(x):K→K§α7→anαn+an−1αn−1+···+a1α+a0=f(α)§ù¼ê¡�Kþ