一．給出五次單位根，輻角(0到2*pi)從小到大

f1=(-1+sqrt(5)+sqrt(-10-2*sqrt(5)))/4; f2=(-1-sqrt(5)+sqrt(-10+2*sqrt(5)))/4;

f3=(-1-sqrt(5)-sqrt(-10+2*sqrt(5)))/4; f4=(-1+sqrt(5)-sqrt(-10-2*sqrt(5)))/4;

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二．求cos(2*k*pi/41)

1.分離實數(shù)項和4個5次根式項

I.求實數(shù)項A，加粗的是實數(shù)項。以下，A110，A120，A210，A220都是實數(shù)，Amnx（x不等于0）都是純的五次根式。以10（8次非5次剩余）和3（5次非2次剩余）為底數(shù)分組。

A110=2*(cos(2*pi/41)+cos(20*pi/41)+cos(36*pi/41)+cos(32*pi/41)+cos(74*pi/41))=(-1+sqrt(41)-sqrt(82-10*sqrt(41)))/4;

A111=2*(cos(2*pi/41)+f1*cos(20*pi/41)+f2*cos(36*pi/41)+f3*cos(32*pi/41)+f4*cos(74*pi/41));

A112=2*(cos(2*pi/41)+f2*cos(20*pi/41)+f4*cos(36*pi/41)+f1*cos(32*pi/41)+f3*cos(74*pi/41));

A113=2*(cos(2*pi/41)+f3*cos(20*pi/41)+f1*cos(36*pi/41)+f4*cos(32*pi/41)+f2*cos(74*pi/41));

A114=2*(cos(2*pi/41)+f4*cos(20*pi/41)+f3*cos(36*pi/41)+f2*cos(32*pi/41)+f1*cos(74*pi/41));

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A120=2*(cos(18*pi/41)+cos(16*pi/41)+cos(78*pi/41)+cos(42*pi/41)+cos(10*pi/41))=(-1+sqrt(41)+sqrt(82-10*sqrt(41)))/4;

A121=2*(cos(18*pi/41)+f1*cos(16*pi/41)+f2*cos(78*pi/41)+f3*cos(42*pi/41)+f4*cos(10*pi/41));

A122=2*(cos(18*pi/41)+f2*cos(16*pi/41)+f4*cos(78*pi/41)+f1*cos(42*pi/41)+f3*cos(10*pi/41));

A123=2*(cos(18*pi/41)+f3*cos(16*pi/41)+f1*cos(78*pi/41)+f4*cos(42*pi/41)+f2*cos(10*pi/41));

A124=2*(cos(18*pi/41)+f4*cos(16*pi/41)+f3*cos(78*pi/41)+f2*cos(42*pi/41)+f1*cos(10*pi/41));

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A210=2*(cos(6*pi/41)+cos(60*pi/41)+cos(26*pi/41)+cos(14*pi/41)+cos(58*pi/41))=(-1-sqrt(41)+sqrt(82+10*sqrt(41)))/4;

A211=2*(cos(6*pi/41)+f1*cos(60*pi/41)+f2*cos(26*pi/41)+f3*cos(14*pi/41)+f4*cos(58*pi/41));

A212=2*(cos(6*pi/41)+f2*cos(60*pi/41)+f4*cos(26*pi/41)+f1*cos(14*pi/41)+f3*cos(58*pi/41));

A213=2*(cos(6*pi/41)+f3*cos(60*pi/41)+f1*cos(26*pi/41)+f4*cos(14*pi/41)+f2*cos(58*pi/41));

A214=2*(cos(6*pi/41)+f4*cos(60*pi/41)+f3*cos(26*pi/41)+f2*cos(14*pi/41)+f1*cos(58*pi/41));

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A220=2*(cos(28*pi/41)+cos(34*pi/41)+cos(12*pi/41)+cos(38*pi/41)+cos(52*pi/41))=(-1-sqrt(41)-sqrt(82+10*sqrt(41)))/4;

A221=2*(cos(28*pi/41)+f1*cos(34*pi/41)+f2*cos(12*pi/41)+f3*cos(38*pi/41)+f4*cos(52*pi/41));

A222=2*(cos(28*pi/41)+f2*cos(34*pi/41)+f4*cos(12*pi/41)+f1*cos(38*pi/41)+f3*cos(52*pi/41));

A223=2*(cos(28*pi/41)+f3*cos(34*pi/41)+f1*cos(12*pi/41)+f4*cos(38*pi/41)+f2*cos(52*pi/41));

A224=2*(cos(28*pi/41)+f4*cos(34*pi/41)+f3*cos(12*pi/41)+f2*cos(38*pi/41)+f1*cos(52*pi/41));

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II.五次根式內(nèi)的表達式G，G的最終表達式加粗。按k是不是41的2次剩余進行分揀整合。

G111=A111^5; G112=A112^5; G113=A113^5; G114=A114^5; G110=A110^5;

G121=A121^5; G122=A122^5; G123=A123^5; G124=A124^5; G120=A120^5;

G211=A211^5; G212=A212^5; G213=A213^5; G214=A214^5; G210=A210^5;

G221=A221^5; G222=A222^5; G223=A223^5; G224=A224^5; G220=A220^5;

引入U，V，W，X，Y五個中間變量。

U11=(G111+G112+G113+G114+G110)/5=-0.25*(1016+14*sqrt(41)-(786-112*sqrt(41))*sqrt(205+32*sqrt(41)));

U12=(G121+G122+G123+G124+G120)/5=-0.25*(1016+14*sqrt(41)+(786-112*sqrt(41))*sqrt(205+32*sqrt(41)));

U21=(G211+G212+G213+G214+G210)/5=-0.25*(1016-14*sqrt(41)-(786+112*sqrt(41))*sqrt(205-32*sqrt(41)));

U22=(G221+G222+G223+G224+G220)/5=-0.25*(1016-14*sqrt(41)+(786+112*sqrt(41))*sqrt(205-32*sqrt(41)));

V11=(G111/f1+G112/f2+G113/f3+G114/f4+G110)/5=-0.25*(2000+120*sqrt(41)-(2320-400*sqrt(41))*sqrt(205+32*sqrt(41)));

V12=(G121/f1+G122/f2+G123/f3+G124/f4+G120)/5=-0.25*(2000+120*sqrt(41)+(2320-400*sqrt(41))*sqrt(205+32*sqrt(41)));

V21=(G211/f1+G212/f2+G213/f3+G214/f4+G210)/5=-0.25*(2000-120*sqrt(41)-(2320+400*sqrt(41))*sqrt(205-32*sqrt(41)));

V22=(G221/f1+G222/f2+G223/f3+G224/f4+G220)/5=-0.25*(2000-120*sqrt(41)+(2320+400*sqrt(41))*sqrt(205-32*sqrt(41)));

W11=(G111/f2+G112/f4+G113/f1+G114/f3+G110)/5=-0.25*(2000+140*sqrt(41)-(1560-260*sqrt(41))*sqrt(205+32*sqrt(41)));

W12=(G121/f2+G122/f4+G123/f1+G124/f3+G120)/5=-0.25*(2000+140*sqrt(41)+(1560-260*sqrt(41))*sqrt(205+32*sqrt(41)));

W21=(G211/f2+G212/f4+G213/f1+G214/f3+G210)/5=-0.25*(2000-140*sqrt(41)-(1560+260*sqrt(41))*sqrt(205-32*sqrt(41)));

W22=(G221/f2+G222/f4+G223/f1+G224/f3+G220)/5=-0.25*(2000-140*sqrt(41)+(1560+260*sqrt(41))*sqrt(205-32*sqrt(41)));

X11=(G111/f3+G112/f1+G113/f4+G114/f2+G110)/5=-0.25*(-460-230*sqrt(41)-(1410-200*sqrt(41))*sqrt(205+32*sqrt(41)));

X12=(G121/f3+G122/f1+G123/f4+G124/f2+G120)/5=-0.25*(-460-230*sqrt(41)+(1410-200*sqrt(41))*sqrt(205+32*sqrt(41)));

X21=(G211/f3+G212/f1+G213/f4+G214/f2+G210)/5=-0.25*(-460+230*sqrt(41)-(1410+200*sqrt(41))*sqrt(205-32*sqrt(41)));

X22=(G221/f3+G222/f1+G223/f4+G224/f2+G220)/5=-0.25*(-460+230*sqrt(41)+(1410+200*sqrt(41))*sqrt(205-32*sqrt(41)));

Y11=(G111/f4+G112/f3+G113/f2+G114/f1+G110)/5=-0.25*(-1895-485*sqrt(41)-(2605-385*sqrt(41))*sqrt(205+32*sqrt(41)));

Y12=(G121/f4+G122/f3+G123/f2+G124/f1+G120)/5=-0.25*(-1895-485*sqrt(41)+(2605-385*sqrt(41))*sqrt(205+32*sqrt(41)));

Y21=(G211/f4+G212/f3+G213/f2+G214/f1+G210)/5=-0.25*(-1895+485*sqrt(41)-(2605+385*sqrt(41))*sqrt(205-32*sqrt(41)));

Y22=(G221/f4+G222/f3+G223/f2+G224/f1+G220)/5=-0.25*(-1895+485*sqrt(41)+(2605+385*sqrt(41))*sqrt(205-32*sqrt(41)));

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整理得到：

G111=-0.25*(2911+499*sqrt(41)+(1819-273*sqrt(41))*sqrt(205+32*sqrt(41))+(3895+605*sqrt(41)+(285+15*sqrt(41))*sqrt(205+32*sqrt(41)))*f1+(3895+625*sqrt(41)+(1045-125*sqrt(41))*sqrt(205+32*sqrt(41)))*f2+(1435+255*sqrt(41)+(1195-185*sqrt(41))*sqrt(205+32*sqrt(41)))*f3);

G112=-0.25*(2911+499*sqrt(41)+(1819-273*sqrt(41))*sqrt(205+32*sqrt(41))+(3895+605*sqrt(41)+(285+15*sqrt(41))*sqrt(205+32*sqrt(41)))*f2+(3895+625*sqrt(41)+(1045-125*sqrt(41))*sqrt(205+32*sqrt(41)))*f4+(1435+255*sqrt(41)+(1195-185*sqrt(41))*sqrt(205+32*sqrt(41)))*f1);

G113=-0.25*(2911+499*sqrt(41)+(1819-273*sqrt(41))*sqrt(205+32*sqrt(41))+(3895+605*sqrt(41)+(285+15*sqrt(41))*sqrt(205+32*sqrt(41)))*f3+(3895+625*sqrt(41)+(1045-125*sqrt(41))*sqrt(205+32*sqrt(41)))*f1+(1435+255*sqrt(41)+(1195-185*sqrt(41))*sqrt(205+32*sqrt(41)))*f4);

G114=-0.25*(2911+499*sqrt(41)+(1819-273*sqrt(41))*sqrt(205+32*sqrt(41))+(3895+605*sqrt(41)+(285+15*sqrt(41))*sqrt(205+32*sqrt(41)))*f4+(3895+625*sqrt(41)+(1045-125*sqrt(41))*sqrt(205+32*sqrt(41)))*f3+(1435+255*sqrt(41)+(1195-185*sqrt(41))*sqrt(205+32*sqrt(41)))*f2);

G121=-0.25*(2911+499*sqrt(41)-(1819-273*sqrt(41))*sqrt(205+32*sqrt(41))+(3895+605*sqrt(41)-(285+15*sqrt(41))*sqrt(205+32*sqrt(41)))*f1+(3895+625*sqrt(41)-(1045-125*sqrt(41))*sqrt(205+32*sqrt(41)))*f2+(1435+255*sqrt(41)-(1195-185*sqrt(41))*sqrt(205+32*sqrt(41)))*f3);

G122=-0.25*(2911+499*sqrt(41)-(1819-273*sqrt(41))*sqrt(205+32*sqrt(41))+(3895+605*sqrt(41)-(285+15*sqrt(41))*sqrt(205+32*sqrt(41)))*f2+(3895+625*sqrt(41)-(1045-125*sqrt(41))*sqrt(205+32*sqrt(41)))*f4+(1435+255*sqrt(41)-(1195-185*sqrt(41))*sqrt(205+32*sqrt(41)))*f1);

G123=-0.25*(2911+499*sqrt(41)-(1819-273*sqrt(41))*sqrt(205+32*sqrt(41))+(3895+605*sqrt(41)-(285+15*sqrt(41))*sqrt(205+32*sqrt(41)))*f3+(3895+625*sqrt(41)-(1045-125*sqrt(41))*sqrt(205+32*sqrt(41)))*f1+(1435+255*sqrt(41)-(1195-185*sqrt(41))*sqrt(205+32*sqrt(41)))*f4);

G124=-0.25*(2911+499*sqrt(41)-(1819-273*sqrt(41))*sqrt(205+32*sqrt(41))+(3895+605*sqrt(41)-(285+15*sqrt(41))*sqrt(205+32*sqrt(41)))*f4+(3895+625*sqrt(41)-(1045-125*sqrt(41))*sqrt(205+32*sqrt(41)))*f3+(1435+255*sqrt(41)-(1195-185*sqrt(41))*sqrt(205+32*sqrt(41)))*f2);

G211=-0.25*(2911-499*sqrt(41)+(1819+273*sqrt(41))*sqrt(205-32*sqrt(41))+(3895-605*sqrt(41)+(285-15*sqrt(41))*sqrt(205-32*sqrt(41)))*f1+(3895-625*sqrt(41)+(1045+125*sqrt(41))*sqrt(205-32*sqrt(41)))*f2+(1435-255*sqrt(41)+(1195+185*sqrt(41))*sqrt(205-32*sqrt(41)))*f3);

G212=-0.25*(2911-499*sqrt(41)+(1819+273*sqrt(41))*sqrt(205-32*sqrt(41))+(3895-605*sqrt(41)+(285-15*sqrt(41))*sqrt(205-32*sqrt(41)))*f2+(3895-625*sqrt(41)+(1045+125*sqrt(41))*sqrt(205-32*sqrt(41)))*f4+(1435-255*sqrt(41)+(1195+185*sqrt(41))*sqrt(205-32*sqrt(41)))*f1);

G213=-0.25*(2911-499*sqrt(41)+(1819+273*sqrt(41))*sqrt(205-32*sqrt(41))+(3895-605*sqrt(41)+(285-15*sqrt(41))*sqrt(205-32*sqrt(41)))*f3+(3895-625*sqrt(41)+(1045+125*sqrt(41))*sqrt(205-32*sqrt(41)))*f1+(1435-255*sqrt(41)+(1195+185*sqrt(41))*sqrt(205-32*sqrt(41)))*f4);

G214=-0.25*(2911-499*sqrt(41)+(1819+273*sqrt(41))*sqrt(205-32*sqrt(41))+(3895-605*sqrt(41)+(285-15*sqrt(41))*sqrt(205-32*sqrt(41)))*f4+(3895-625*sqrt(41)+(1045+125*sqrt(41))*sqrt(205-32*sqrt(41)))*f3+(1435-255*sqrt(41)+(1195+185*sqrt(41))*sqrt(205-32*sqrt(41)))*f2);

G221=-0.25*(2911-499*sqrt(41)-(1819+273*sqrt(41))*sqrt(205-32*sqrt(41))+(3895-605*sqrt(41)-(285-15*sqrt(41))*sqrt(205-32*sqrt(41)))*f1+(3895-625*sqrt(41)-(1045+125*sqrt(41))*sqrt(205-32*sqrt(41)))*f2+(1435-255*sqrt(41)-(1195+185*sqrt(41))*sqrt(205-32*sqrt(41)))*f3);

G222=-0.25*(2911-499*sqrt(41)-(1819+273*sqrt(41))*sqrt(205-32*sqrt(41))+(3895-605*sqrt(41)-(285-15*sqrt(41))*sqrt(205-32*sqrt(41)))*f2+(3895-625*sqrt(41)-(1045+125*sqrt(41))*sqrt(205-32*sqrt(41)))*f4+(1435-255*sqrt(41)-(1195+185*sqrt(41))*sqrt(205-32*sqrt(41)))*f1);

G223=-0.25*(2911-499*sqrt(41)-(1819+273*sqrt(41))*sqrt(205-32*sqrt(41))+(3895-605*sqrt(41)-(285-15*sqrt(41))*sqrt(205-32*sqrt(41)))*f3+(3895-625*sqrt(41)-(1045+125*sqrt(41))*sqrt(205-32*sqrt(41)))*f1+(1435-255*sqrt(41)-(1195+185*sqrt(41))*sqrt(205-32*sqrt(41)))*f4);

G224=-0.25*(2911-499*sqrt(41)-(1819+273*sqrt(41))*sqrt(205-32*sqrt(41))+(3895-605*sqrt(41)-(285-15*sqrt(41))*sqrt(205-32*sqrt(41)))*f4+(3895-625*sqrt(41)-(1045+125*sqrt(41))*sqrt(205-32*sqrt(41)))*f3+(1435-255*sqrt(41)-(1195+185*sqrt(41))*sqrt(205-32*sqrt(41)))*f2);

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2.比對五次根式的輻角，并用A、G^(1/5)線性表示cos(2*k*pi/41)。

代入已知數(shù)值得到：

II.cos(2*k*pi/41)，k是41的2次剩余：

cos(2*pi/41)=(A110+G111^(1/5)+G114^(1/5)+f3*G112^(1/5)+f2*G113^(1/5))/10;

cos(20*pi/41)=(A110+f4*G111^(1/5)+f1*G114^(1/5)+f1*G112^(1/5)+f4*G113^(1/5))/10;

cos(36*pi/41)=(A110+f3*G111^(1/5)+f2*G114^(1/5)+f4*G112^(1/5)+f1*G113^(1/5))/10;

cos(32*pi/41)=(A110+f2*G111^(1/5)+f3*G114^(1/5)+f2*G112^(1/5)+f3*G113^(1/5))/10;

cos(74*pi/41)=(A110+f1*G111^(1/5)+f4*G114^(1/5)+G112^(1/5)+G113^(1/5))/10;

cos(18*pi/41)=(A120+f1*G121^(1/5)+f4*G124^(1/5)+f3*G122^(1/5)+f2*G123^(1/5))/10;

cos(16*pi/41)=(A120+G121^(1/5)+G124^(1/5)+f1*G122^(1/5)+f4*G123^(1/5))/10;

cos(78*pi/41)=(A120+f4*G121^(1/5)+f1*G124^(1/5)+f4*G122^(1/5)+f1*G123^(1/5))/10;

cos(42*pi/41)=(A120+f3*G121^(1/5)+f2*G124^(1/5)+f2*G122^(1/5)+f3*G123^(1/5))/10;

cos(10*pi/41)=(A120+f2*G121^(1/5)+f3*G124^(1/5)+G122^(1/5)+G123^(1/5))/10;

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II.cos(2*k*pi/41)，k不是41的2次剩余：

cos(6*pi/41)=(A210+G211^(1/5)+G214^(1/5)+f1*G212^(1/5)+f4*G213^(1/5))/10;

cos(60*pi/41)=(A210+f4*G211^(1/5)+f1*G214^(1/5)+f4*G212^(1/5)+f1*G213^(1/5))/10;

cos(26*pi/41)=(A210+f3*G211^(1/5)+f2*G214^(1/5)+f2*G212^(1/5)+f3*G213^(1/5))/10;

cos(14*pi/41)=(A210+f2*G211^(1/5)+f3*G214^(1/5)+G212^(1/5)+G213^(1/5))/10;

cos(58*pi/41)=(A210+f1*G211^(1/5)+f4*G214^(1/5)+f3*G212^(1/5)+f2*G213^(1/5))/10;

cos(28*pi/41)=(A220+f2*G221^(1/5)+f3*G224^(1/5)+f4*G222^(1/5)+f1*G223^(1/5))/10;

cos(34*pi/41)=(A220+f1*G221^(1/5)+f4*G224^(1/5)+f2*G222^(1/5)+f3*G223^(1/5))/10;

cos(12*pi/41)=(A220+G221^(1/5)+G224^(1/5)+G222^(1/5)+G223^(1/5))/10;（很特殊，五次根號外面沒有任何的五次根式系數(shù)。因為A220實在是太負了）

cos(38*pi/41)=(A220+f4*G221^(1/5)+f1*G224^(1/5)+f3*G222^(1/5)+f2*G223^(1/5))/10;

cos(52*pi/41)=(A220+f3*G221^(1/5)+f2*G224^(1/5)+f1*G222^(1/5)+f4*G223^(1/5))/10.

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（未完待續(xù)，下一期，求sin(2*k*pi/41)完整解析式）