一．五次和三次單位根

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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w1=(-1+sqrt(-3))/2; w2=(-1-sqrt(-3))/2.

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二．求解cos(2*k*pi/31)

1.提取不含5次根號的實(shí)數(shù)項(xiàng)Ax1，是同一個整系數(shù)三次方程的三個不同實(shí)數(shù)根。

先以5為底數(shù)解三次方程、后以2為底數(shù)分組解五次方程。

A11=2*(cos(2*pi/31)+cos(4*pi/31)+cos(8*pi/31)+cos(16*pi/31)+cos(32*pi/31))，是實(shí)數(shù);

A12=2*(cos(2*pi/31)+f1*cos(4*pi/31)+f2*cos(8*pi/31)+f3*cos(16*pi/31)+f4*cos(32*pi/31));

A13=2*(cos(2*pi/31)+f2*cos(4*pi/31)+f4*cos(8*pi/31)+f1*cos(16*pi/31)+f3*cos(32*pi/31));

A14=2*(cos(2*pi/31)+f3*cos(4*pi/31)+f1*cos(8*pi/31)+f4*cos(16*pi/31)+f2*cos(32*pi/31));

A15=2*(cos(2*pi/31)+f4*cos(4*pi/31)+f3*cos(8*pi/31)+f2*cos(16*pi/31)+f1*cos(32*pi/31));

A21=2*(cos(10*pi/31)+cos(20*pi/31)+cos(40*pi/31)+cos(18*pi/31)+cos(36*pi/31))，是實(shí)數(shù);

A22=2*(cos(10*pi/31)+f1*cos(20*pi/31)+f2*cos(40*pi/31)+f3*cos(18*pi/31)+f4*cos(36*pi/31));

A23=2*(cos(10*pi/31)+f2*cos(20*pi/31)+f4*cos(40*pi/31)+f1*cos(18*pi/31)+f3*cos(36*pi/31));

A24=2*(cos(10*pi/31)+f3*cos(20*pi/31)+f1*cos(40*pi/31)+f4*cos(18*pi/31)+f2*cos(36*pi/31));

A25=2*(cos(10*pi/31)+f4*cos(20*pi/31)+f3*cos(40*pi/31)+f2*cos(18*pi/31)+f1*cos(36*pi/31));

A31=2*(cos(50*pi/31)+cos(38*pi/31)+cos(14*pi/31)+cos(28*pi/31)+cos(56*pi/31))，是實(shí)數(shù);

A32=2*(cos(50*pi/31)+f1*cos(38*pi/31)+f2*cos(14*pi/31)+f3*cos(28*pi/31)+f4*cos(56*pi/31));

A33=2*(cos(50*pi/31)+f2*cos(38*pi/31)+f4*cos(14*pi/31)+f1*cos(28*pi/31)+f3*cos(56*pi/31));

A34=2*(cos(50*pi/31)+f3*cos(38*pi/31)+f1*cos(14*pi/31)+f4*cos(28*pi/31)+f2*cos(56*pi/31));

A35=2*(cos(50*pi/31)+f4*cos(38*pi/31)+f3*cos(14*pi/31)+f2*cos(28*pi/31)+f1*cos(56*pi/31));

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A11+A21+A31=-1;

(A11+w1*A21+w2*A31)^3=62-93*sqrt(-3);

(A11+w2*A21+w1*A31)^3=62+93*sqrt(-3);

那么

A11=(-1+(62+93*sqrt(-3))^(1/3)+(62-93*sqrt(-3))^(1/3))/3;

A21=(-1+w1*(62+93*sqrt(-3))^(1/3)+w2*(62-93*sqrt(-3))^(1/3))/3;

A31=(-1+w2*(62+93*sqrt(-3))^(1/3)+w1*(62-93*sqrt(-3))^(1/3))/3;

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G11=A11^5; G12=A12^5; G13=A13^5; G14=A14^5; G15=A15^5;

G21=A21^5; G22=A22^5; G23=A23^5; G24=A24^5; G25=A25^5;

G31=A31^5; G32=A32^5; G33=A33^5; G34=A34^5; G35=A35^5;

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2.設(shè)五次根式內(nèi)Gxy=Jx1+Jx2*fy-1+Jx3*(fy-1)^2+Jx4*(fy-1)^3+Jx5*(fy-1)^4。J全部滿足整系數(shù)的一元三次方程！求解對應(yīng)的5個三次方程。

2-1第一項(xiàng)

J11=(G11+G12+G13+G14+G15)/5;

J21=(G21+G22+G23+G24+G25)/5;

J31=(G31+G32+G33+G34+G35)/5;

J11=(604+(137-22*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(137+22*sqrt(-3))*(62-93*sqrt(-3))^(1/3))/3;

J21=(604+(137-22*sqrt(-3))*w1*(62+93*sqrt(-3))^(1/3)+(137+22*sqrt(-3))*w2*(62-93*sqrt(-3))^(1/3))/3;

J31=(604+(137-22*sqrt(-3))*w2*(62+93*sqrt(-3))^(1/3)+(137+22*sqrt(-3))*w1*(62-93*sqrt(-3))^(1/3))/3;

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2-2第二項(xiàng)

J12=(G11+G12/f1+G13/f2+G14/f3+G15/f4)/5;

J22=(G21+G22/f1+G23/f2+G24/f3+G25/f4)/5;

J32=(G31+G32/f1+G33/f2+G34/f3+G35/f4)/5;

J12=(480+(127.5-57.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(127.5+57.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))/3;

J22=(480+(127.5-57.5*sqrt(-3))*w1*(62+93*sqrt(-3))^(1/3)+(127.5+57.5*sqrt(-3))*w2*(62-93*sqrt(-3))^(1/3))/3;

J32=(480+(127.5-57.5*sqrt(-3))*w2*(62+93*sqrt(-3))^(1/3)+(127.5+57.5*sqrt(-3))*w1*(62-93*sqrt(-3))^(1/3))/3;

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2-3第三項(xiàng)

J13=(G11+G12/f2+G13/f4+G14/f1+G15/f3)/5;

J23=(G21+G22/f2+G23/f4+G24/f1+G25/f3)/5;

J33=(G31+G32/f2+G33/f4+G34/f1+G35/f3)/5;

J13=(15+(-30+65*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-30-65*sqrt(-3))*(62-93*sqrt(-3))^(1/3))/3;

J23=(15+(-30+65*sqrt(-3))*w1*(62+93*sqrt(-3))^(1/3)+(-30-65*sqrt(-3))*w2*(62-93*sqrt(-3))^(1/3))/3;

J33=(15+(-30+65*sqrt(-3))*w2*(62+93*sqrt(-3))^(1/3)+(-30-65*sqrt(-3))*w1*(62-93*sqrt(-3))^(1/3))/3;

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2-4第四項(xiàng)

J14=(G11+G12/f3+G13/f1+G14/f4+G15/f2)/5;

J24=(G21+G22/f3+G23/f1+G24/f4+G25/f2)/5;

J34=(G31+G32/f3+G33/f1+G34/f4+G35/f2)/5;

J14=(-1070+(-107.5-2.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-107.5+2.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))/3;

J24=(-1070+(-107.5-2.5*sqrt(-3))*w1*(62+93*sqrt(-3))^(1/3)+(-107.5+2.5*sqrt(-3))*w2*(62-93*sqrt(-3))^(1/3))/3;

J34=(-1070+(-107.5-2.5*sqrt(-3))*w2*(62+93*sqrt(-3))^(1/3)+(-107.5+2.5*sqrt(-3))*w1*(62-93*sqrt(-3))^(1/3))/3;

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2-5第五項(xiàng)

J15=(G11+G12/f4+G13/f3+G14/f2+G15/f1)/5;

J25=(G21+G22/f4+G23/f3+G24/f2+G25/f1)/5;

J35=(G31+G32/f4+G33/f3+G34/f2+G35/f1)/5;

J15=(-140+(-25+30*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-25-30*sqrt(-3))*(62-93*sqrt(-3))^(1/3))/3;

J25=(-140+(-25+30*sqrt(-3))*w1*(62+93*sqrt(-3))^(1/3)+(-25-30*sqrt(-3))*w2*(62-93*sqrt(-3))^(1/3))/3;

J35=(-140+(-25+30*sqrt(-3))*w2*(62+93*sqrt(-3))^(1/3)+(-25-30*sqrt(-3))*w1*(62-93*sqrt(-3))^(1/3))/3;

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2-6組裝得到G值

代入中間變量J以及根據(jù)1+f1+f2+f3+f4=0：

G12=(744+(162-52*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(162+52*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(152.5-87.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(152.5+87.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1+(155+(-5+35*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-5-35*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2-(930+(82.5+32.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(82.5-32.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3)/3;

G13=(744+(162-52*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(162+52*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(152.5-87.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(152.5+87.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2+(155+(-5+35*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-5-35*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4-(930+(82.5+32.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(82.5-32.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1)/3;

G14=(744+(162-52*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(162+52*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(152.5-87.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(152.5+87.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3+(155+(-5+35*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-5-35*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1-(930+(82.5+32.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(82.5-32.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4)/3;

G15=(744+(162-52*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(162+52*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(152.5-87.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(152.5+87.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4+(155+(-5+35*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-5-35*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3-(930+(82.5+32.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(82.5-32.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2)/3;

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G22=(744+(-3+107*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-3-107*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(55+120*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(55-120*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1+(155+(-50-20*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-50+20*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2-(930+(-90+25*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-90-25*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3)/3;

G23=(744+(-3+107*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-3-107*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(55+120*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(55-120*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2+(155+(-50-20*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-50+20*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4-(930+(-90+25*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-90-25*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1)/3;

G24=(744+(-3+107*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-3-107*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(55+120*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(55-120*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3+(155+(-50-20*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-50+20*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1-(930+(-90+25*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-90-25*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4)/3;

G25=(744+(-3+107*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-3-107*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(55+120*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(55-120*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4+(155+(-50-20*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-50+20*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3-(930+(-90+25*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-90-25*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2)/3;

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G32=(744+(-159-55*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-159+55*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(-207.5-32.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-207.5+32.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1+(155+(55-15*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(55+15*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2-(930+(7.5-57.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(7.5+57.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3)/3;

G33=(744+(-159-55*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-159+55*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(-207.5-32.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-207.5+32.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2+(155+(55-15*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(55+15*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4-(930+(7.5-57.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(7.5+57.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1)/3;

G34=(744+(-159-55*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-159+55*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(-207.5-32.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-207.5+32.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3+(155+(55-15*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(55+15*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f1-(930+(7.5-57.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(7.5+57.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4)/3;

G35=(744+(-159-55*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-159+55*sqrt(-3))*(62-93*sqrt(-3))^(1/3)+(620+(-207.5-32.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(-207.5+32.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f4+(155+(55-15*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(55+15*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f3-(930+(7.5-57.5*sqrt(-3))*(62+93*sqrt(-3))^(1/3)+(7.5+57.5*sqrt(-3))*(62-93*sqrt(-3))^(1/3))*f2)/3;

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三．比對五次根式G^(1/5)和Axy（y不等于1）的輻角，并用Ax1、G^(1/5)線性表示cos(2*k*pi/31)。上述參數(shù)都有加粗的具體表達(dá)式。

3-1 k=1為開頭，2為底數(shù)

cos(2*pi/31)=(A11+f1*G12^0.2+f4*G15^0.2+G13^0.2+G14^0.2)/10;

cos(4*pi/31)=(A11+G12^0.2+G15^0.2+f3*G13^0.2+f2*G14^0.2)/10;

cos(8*pi/31)=(A11+f4*G12^0.2+f1*G15^0.2+f1*G13^0.2+f4*G14^0.2)/10;

cos(16*pi/31)=(A11+f3*G12^0.2+f2*G15^0.2+f4*G13^0.2+f1*G14^0.2)/10;

cos(32*pi/31)=(A11+f2*G12^0.2+f3*G15^0.2+f2*G13^0.2+f3*G14^0.2)/10;

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3-2 k=5為開頭，2為底數(shù)

cos(10*pi/31)=(A21+G22^0.2+G25^0.2+G23^0.2+G24^0.2)/10;（這是因?yàn)锳21太負(fù)了）

cos(20*pi/31)=(A21+f4*G22^0.2+f1*G25^0.2+f3*G23^0.2+f2*G24^0.2)/10;

cos(40*pi/31)=(A21+f3*G22^0.2+f2*G25^0.2+f1*G23^0.2+f4*G24^0.2)/10;

cos(18*pi/31)=(A21+f2*G22^0.2+f3*G25^0.2+f4*G23^0.2+f1*G24^0.2)/10;

cos(36*pi/31)=(A21+f1*G22^0.2+f4*G25^0.2+f2*G23^0.2+f3*G24^0.2)/10;

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3-3 k=25為開頭，2為底數(shù)

cos(50*pi/31)=(A31+f4*G32^0.2+f1*G35^0.2+f4*G33^0.2+f1*G34^0.2)/10;

cos(38*pi/31)=(A31+f3*G32^0.2+f2*G35^0.2+f2*G33^0.2+f3*G34^0.2)/10;

cos(14*pi/31)=(A31+f2*G32^0.2+f3*G35^0.2+G33^0.2+G34^0.2)/10;

cos(28*pi/31)=(A31+f1*G32^0.2+f4*G35^0.2+f3*G33^0.2+f2*G34^0.2)/10;

cos(56*pi/31)=(A31+G32^0.2+G35^0.2+f1*G33^0.2+f4*G34^0.2)/10.