三．求sin(2*k*pi/41)

I.求實數(shù)項B。以下，B110，B120，B210，B220都是實數(shù)，Bmnx（x不等于0）都是純的五次根式。前一章已經聲明五次單位根f1~f4的值。

B110=2*(sin(2*pi/41)+sin(20*pi/41)+sin(36*pi/41)+sin(32*pi/41)+sin(74*pi/41))=sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))/2;

B111=2*(sin(2*pi/41)+f1*sin(20*pi/41)+f2*sin(36*pi/41)+f3*sin(32*pi/41)+f4*sin(74*pi/41));

B112=2*(sin(2*pi/41)+f2*sin(20*pi/41)+f4*sin(36*pi/41)+f1*sin(32*pi/41)+f3*sin(74*pi/41));

B113=2*(sin(2*pi/41)+f3*sin(20*pi/41)+f1*sin(36*pi/41)+f4*sin(32*pi/41)+f2*sin(74*pi/41));

B114=2*(sin(2*pi/41)+f4*sin(20*pi/41)+f3*sin(36*pi/41)+f2*sin(32*pi/41)+f1*sin(74*pi/41));

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B120=2*(sin(18*pi/41)+sin(16*pi/41)+sin(78*pi/41)+sin(42*pi/41)+sin(10*pi/41))=sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))/2;

B121=2*(sin(18*pi/41)+f1*sin(16*pi/41)+f2*sin(78*pi/41)+f3*sin(42*pi/41)+f4*sin(10*pi/41));

B122=2*(sin(18*pi/41)+f2*sin(16*pi/41)+f4*sin(78*pi/41)+f1*sin(42*pi/41)+f3*sin(10*pi/41));

B123=2*(sin(18*pi/41)+f3*sin(16*pi/41)+f1*sin(78*pi/41)+f4*sin(42*pi/41)+f2*sin(10*pi/41));

B124=2*(sin(18*pi/41)+f4*sin(16*pi/41)+f3*sin(78*pi/41)+f2*sin(42*pi/41)+f1*sin(10*pi/41));

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B210=2*(sin(6*pi/41)+sin(60*pi/41)+sin(26*pi/41)+sin(14*pi/41)+sin(58*pi/41))=sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))/2;

B211=2*(sin(6*pi/41)+f1*sin(60*pi/41)+f2*sin(26*pi/41)+f3*sin(14*pi/41)+f4*sin(58*pi/41));

B212=2*(sin(6*pi/41)+f2*sin(60*pi/41)+f4*sin(26*pi/41)+f1*sin(14*pi/41)+f3*sin(58*pi/41));

B213=2*(sin(6*pi/41)+f3*sin(60*pi/41)+f1*sin(26*pi/41)+f4*sin(14*pi/41)+f2*sin(58*pi/41));

B214=2*(sin(6*pi/41)+f4*sin(60*pi/41)+f3*sin(26*pi/41)+f2*sin(14*pi/41)+f1*sin(58*pi/41));

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B220=2*(sin(28*pi/41)+sin(34*pi/41)+sin(12*pi/41)+sin(38*pi/41)+sin(52*pi/41))=sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))/2;

B221=2*(sin(28*pi/41)+f1*sin(34*pi/41)+f2*sin(12*pi/41)+f3*sin(38*pi/41)+f4*sin(52*pi/41));

B222=2*(sin(28*pi/41)+f2*sin(34*pi/41)+f4*sin(12*pi/41)+f1*sin(38*pi/41)+f3*sin(52*pi/41));

B223=2*(sin(28*pi/41)+f3*sin(34*pi/41)+f1*sin(12*pi/41)+f4*sin(38*pi/41)+f2*sin(52*pi/41));

B224=2*(sin(28*pi/41)+f4*sin(34*pi/41)+f3*sin(12*pi/41)+f2*sin(38*pi/41)+f1*sin(52*pi/41));

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II.求五次根式內的表達式H。

H111=B111^5; H112=B112^5; H113=B113^5; H114=B114^5; H110=B110^5;

H121=B121^5; H122=B122^5; H123=B123^5; H124=B124^5; H120=B120^5;

H211=B211^5; H212=B212^5; H213=B213^5; H214=B214^5; H210=B210^5;

H221=B221^5; H222=B222^5; H223=B223^5; H224=B224^5; H220=B220^5;

為了方便計算，引入中間變量I：Ixyz=Hxyz/Bxy0.

I111=H111/B110; I112=H112/B110; I113=H113/B110; I114=H114/B110; I110=H110/B110;

I121=H121/B120; I122=H122/B120; I123=H123/B120; I124=H124/B120; I120=H120/B120;

I211=H211/B210; I212=H212/B210; I213=H213/B210; I214=H214/B210; I210=H210/B210;

I221=H221/B220; I222=H222/B220; I223=H223/B220; I224=H224/B220; I220=H220/B220;

再引入O、P、Q、R、S五個中間變量：

O11=(I111+I112+I113+I114+I110)/5=0.125*(5095-713*sqrt(41)-(1061+93*sqrt(41))*sqrt(205-32*sqrt(41)));

O12=(I121+I122+I123+I124+I120)/5=0.125*(5095-713*sqrt(41)+(1061+93*sqrt(41))*sqrt(205-32*sqrt(41)));

O21=(I211+I212+I213+I214+I210)/5=0.125*(5095+713*sqrt(41)+(1061-93*sqrt(41))*sqrt(205+32*sqrt(41)));

O22=(I221+I222+I223+I224+I220)/5=0.125*(5095+713*sqrt(41)-(1061-93*sqrt(41))*sqrt(205+32*sqrt(41)));

P11=(I111/f1+I112/f2+I113/f3+I114/f4+I110)/5=0.125*(-4400+760*sqrt(41)-(420+140*sqrt(41))*sqrt(205-32*sqrt(41)));

P12=(I121/f1+I122/f2+I123/f3+I124/f4+I120)/5=0.125*(-4400+760*sqrt(41)+(420+140*sqrt(41))*sqrt(205-32*sqrt(41)));

P21=(I211/f1+I212/f2+I213/f3+I214/f4+I210)/5=0.125*(-4400-760*sqrt(41)+(420-140*sqrt(41))*sqrt(205+32*sqrt(41)));

P22=(I221/f1+I222/f2+I223/f3+I224/f4+I220)/5=0.125*(-4400-760*sqrt(41)-(420-140*sqrt(41))*sqrt(205+32*sqrt(41)));

Q11=(I111/f2+I112/f4+I113/f1+I114/f3+I110)/5=0.125*(2740-380*sqrt(41)-240*sqrt(205-32*sqrt(41)));

Q12=(I121/f2+I122/f4+I123/f1+I124/f3+I120)/5=0.125*(2740-380*sqrt(41)+240*sqrt(205-32*sqrt(41)));

Q21=(I211/f2+I212/f4+I213/f1+I214/f3+I210)/5=0.125*(2740+380*sqrt(41)+240*sqrt(205+32*sqrt(41)));

Q22=(I221/f3+I222/f1+I223/f4+I224/f2+I220)/5=0.125*(2740+380*sqrt(41)-240*sqrt(205+32*sqrt(41)));

R11=(I111/f3+I112/f1+I113/f4+I114/f2+I110)/5=0.125*(1775-225*sqrt(41)+(75+35*sqrt(41))*sqrt(205-32*sqrt(41)));

R12=(I121/f3+I122/f1+I123/f4+I124/f2+I120)/5=0.125*(1775-225*sqrt(41)-(75+35*sqrt(41))*sqrt(205-32*sqrt(41)));

R21=(I211/f3+I212/f1+I213/f4+I214/f2+I210)/5=0.125*(1775+225*sqrt(41)-(75-35*sqrt(41))*sqrt(205+32*sqrt(41)));

R22=(I221/f3+I222/f1+I223/f4+I224/f2+I220)/5=0.125*(1775+225*sqrt(41)+(75-35*sqrt(41))*sqrt(205+32*sqrt(41)));

S11=(I111/f4+I112/f3+I113/f2+I114/f1+I110)/5=0.125*(-3980+680*sqrt(41)-(240+100*sqrt(41))*sqrt(205-32*sqrt(41)));

S12=(I121/f4+I122/f3+I123/f2+I124/f1+I120)/5=0.125*(-3980+680*sqrt(41)+(240+100*sqrt(41))*sqrt(205-32*sqrt(41)));

S21=(I211/f4+I212/f3+I213/f2+I214/f1+I210)/5=0.125*(-3980-680*sqrt(41)+(240-100*sqrt(41))*sqrt(205+32*sqrt(41)));

S22=(I221/f4+I222/f3+I223/f2+I224/f1+I220)/5=0.125*(-3980-680*sqrt(41)-(240-100*sqrt(41))*sqrt(205+32*sqrt(41)));

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根據1+f1+f2+f3+f4=0整理得到：

H111=0.0625*sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)-(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)-(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f1+(6720-1060*sqrt(41)+100*sqrt(8405-1312*sqrt(41)))*f2+(5755-905*sqrt(41)+(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f3);

H112=0.0625*sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)-(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)-(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f2+(6720-1060*sqrt(41)+100*sqrt(8405-1312*sqrt(41)))*f4+(5755-905*sqrt(41)+(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f1);

H113=0.0625*sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)-(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)-(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f3+(6720-1060*sqrt(41)+100*sqrt(8405-1312*sqrt(41)))*f1+(5755-905*sqrt(41)+(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f4);

H114=0.0625*sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)-(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)-(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f4+(6720-1060*sqrt(41)+100*sqrt(8405-1312*sqrt(41)))*f3+(5755-905*sqrt(41)+(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f2);

H121=0.0625*sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)+(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)+(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f1+(6720-1060*sqrt(41)-100*sqrt(8405-1312*sqrt(41)))*f2+(5755-905*sqrt(41)-(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f3);

H122=0.0625*sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)+(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)+(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f2+(6720-1060*sqrt(41)-100*sqrt(8405-1312*sqrt(41)))*f4+(5755-905*sqrt(41)-(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f1);

H123=0.0625*sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)+(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)+(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f3+(6720-1060*sqrt(41)-100*sqrt(8405-1312*sqrt(41)))*f1+(5755-905*sqrt(41)-(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f4);

H124=0.0625*sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)+(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)+(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f4+(6720-1060*sqrt(41)-100*sqrt(8405-1312*sqrt(41)))*f3+(5755-905*sqrt(41)-(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f2);

H211=0.0625*sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)+(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)+(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f1+(6720+1060*sqrt(41)+100*sqrt(8405+1312*sqrt(41)))*f2+(5755+905*sqrt(41)-(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f3);

H212=0.0625*sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)+(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)+(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f2+(6720+1060*sqrt(41)+100*sqrt(8405+1312*sqrt(41)))*f4+(5755+905*sqrt(41)-(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f1);

H213=0.0625*sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)+(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)+(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f3+(6720+1060*sqrt(41)+100*sqrt(8405+1312*sqrt(41)))*f1+(5755+905*sqrt(41)-(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f4);

H214=0.0625*sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)+(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)+(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f4+(6720+1060*sqrt(41)+100*sqrt(8405+1312*sqrt(41)))*f3+(5755+905*sqrt(41)-(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f2);

H221=0.0625*sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)-(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)-(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f1+(6720+1060*sqrt(41)-100*sqrt(8405+1312*sqrt(41)))*f2+(5755+905*sqrt(41)+(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f3);

H222=0.0625*sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)-(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)-(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f2+(6720+1060*sqrt(41)-100*sqrt(8405+1312*sqrt(41)))*f4+(5755+905*sqrt(41)+(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f1);

H223=0.0625*sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)-(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)-(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f3+(6720+1060*sqrt(41)-100*sqrt(8405+1312*sqrt(41)))*f1+(5755+905*sqrt(41)+(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f4);

H224=0.0625*sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)-(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)-(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f4+(6720+1060*sqrt(41)-100*sqrt(8405+1312*sqrt(41)))*f3+(5755+905*sqrt(41)+(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f2);

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2.比對五次根式的輻角，并用B、H^(1/5)線性表示sin(2*k*pi/41)。這些數(shù)值比cos(2*k*pi/41)復雜一些。

代入已知數(shù)值得到：

I.k為2次剩余

sin(2*pi/41)=(B110+f2*H111^(1/5)+f3*H114^(1/5)+f1*H112^(1/5)+f4*H113^(1/5))/10;

sin(20*pi/41)=(B110+f1*H111^(1/5)+f4*H114^(1/5)+f4*H112^(1/5)+f1*H113^(1/5))/10;

sin(36*pi/41)=(B110+H111^(1/5)+H114^(1/5)+f2*H112^(1/5)+f3*H113^(1/5))/10;

sin(32*pi/41)=(B110+f4*H111^(1/5)+f1*H114^(1/5)+H112^(1/5)+H113^(1/5))/10;

sin(74*pi/41)=(B110+f3*H111^(1/5)+f2*H114^(1/5)+f3*H112^(1/5)+f2*H113^(1/5))/10;

sin(18*pi/41)=(B120+H121^(1/5)+H124^(1/5)+f2*H122^(1/5)+f3*H123^(1/5))/10;

sin(16*pi/41)=(B120+f4*H121^(1/5)+f1*H124^(1/5)+H122^(1/5)+H123^(1/5))/10;

sin(78*pi/41)=(B120+f3*H121^(1/5)+f2*H124^(1/5)+f3*H122^(1/5)+f2*H123^(1/5))/10;

sin(42*pi/41)=(B120+f2*H121^(1/5)+f3*H124^(1/5)+f1*H122^(1/5)+f4*H123^(1/5))/10;

sin(10*pi/41)=(B120+f1*H121^(1/5)+f4*H124^(1/5)+f4*H122^(1/5)+f1*H123^(1/5))/10;

II.k不為2次剩余

sin(6*pi/41)=(B210+f3*H211^(1/5)+f2*H214^(1/5)+H212^(1/5)+H213^(1/5))/10;

sin(60*pi/41)=(B210+f2*H211^(1/5)+f3*H214^(1/5)+f3*H212^(1/5)+f2*H213^(1/5))/10;

sin(26*pi/41)=(B210+f1*H211^(1/5)+f4*H214^(1/5)+f1*H212^(1/5)+f4*H213^(1/5))/10;

sin(14*pi/41)=(B210+H211^(1/5)+H214^(1/5)+f4*H212^(1/5)+f1*H213^(1/5))/10;

sin(58*pi/41)=(B210+f4*H211^(1/5)+f1*H214^(1/5)+f2*H212^(1/5)+f3*H213^(1/5))/10;

sin(28*pi/41)=(B220+f1*H221^(1/5)+f4*H224^(1/5)+H222^(1/5)+H223^(1/5))/10;

sin(34*pi/41)=(B220+H221^(1/5)+H224^(1/5)+f3*H222^(1/5)+f2*H223^(1/5))/10;

sin(12*pi/41)=(B220+f4*H221^(1/5)+f1*H224^(1/5)+f1*H222^(1/5)+f4*H223^(1/5))/10;

sin(38*pi/41)=(B220+f3*H221^(1/5)+f2*H224^(1/5)+f4*H222^(1/5)+f1*H223^(1/5))/10;

sin(52*pi/41)=(B220+f2*H221^(1/5)+f3*H224^(1/5)+f2*H222^(1/5)+f3*H223^(1/5))/10;

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四．41次單位根舉例

前一章聲明了數(shù)列A、G的表達式。根據exp(j*x)=cos(x)+j*sin(x)得到：

exp(2*pi*j/41)=(A110+G111^(1/5)+G114^(1/5)+f3*G112^(1/5)+f2*G113^(1/5)+j*(B110+f2*H111^(1/5)+f3*H114^(1/5)+f1*H112^(1/5)+f4*H113^(1/5)))/10;

exp(6*pi*j/41)=(A210+G211^(1/5)+G214^(1/5)+f1*G212^(1/5)+f4*G213^(1/5)+j*(B210+f3*H211^(1/5)+f2*H214^(1/5)+H212^(1/5)+H213^(1/5)))/10;

exp(18*pi*j/41)=(A120+f1*G121^(1/5)+f4*G124^(1/5)+f3*G122^(1/5)+f2*G123^(1/5)+j*(B120+H121^(1/5)+H124^(1/5)+f2*H122^(1/5)+f3*H123^(1/5)))/10;

exp(54*pi*j/41)=(A220+f2*G221^(1/5)+f3*G224^(1/5)+f4*G222^(1/5)+f1*G223^(1/5)-j*(B220+f1*H221^(1/5)+f4*H224^(1/5)+H222^(1/5)+H223^(1/5)))/10.

其他的單位根也很容易得到。

五．寫在最后

這些計算結果表明正41邊形可通過至少兩次5等分角作出?？蓪p主x_4597的x^41-1=0求解視頻內容，作為有益補充。但難免有缺漏不足之處，敬請諸公斧正。

謝謝大家！