43邊形（二）
續(xù)上一篇《43邊形（一）》
三．求解sin(2*k*pi/43)
1.?14*sin(2*k*pi/43)的非7次根式、含三次根號(hào)和因子sqrt(43)的實(shí)數(shù)項(xiàng)
B10=2*(sin(2*pi/43)+sin(82*pi/43)+sin(8*pi/43)+sin(70*pi/43)+sin(32*pi/43)+sin(22*pi/43)+sin(42*pi/43));
B20=2*(sin(12*pi/43)+sin(62*pi/43)+sin(48*pi/43)+sin(76*pi/43)+sin(20*pi/43)+sin(46*pi/43)+sin(80*pi/43));
B30=2*(sin(72*pi/43)+sin(28*pi/43)+sin(30*pi/43)+sin(26*pi/43)+sin(34*pi/43)+sin(18*pi/43)+sin(50*pi/43));
B11=2*(sin(2*pi/43)+v1*sin(82*pi/43)+v2*sin(8*pi/43)+v3*sin(70*pi/43)+v4*sin(32*pi/43)+v5*sin(22*pi/43)+v6*sin(42*pi/43));
B21=2*(sin(12*pi/43)+v1*sin(62*pi/43)+v2*sin(48*pi/43)+v3*sin(76*pi/43)+v4*sin(20*pi/43)+v5*sin(46*pi/43)+v6*sin(80*pi/43));
B31=2*(sin(72*pi/43)+v1*sin(28*pi/43)+v2*sin(30*pi/43)+v3*sin(26*pi/43)+v4*sin(34*pi/43)+v5*sin(18*pi/43)+v6*sin(50*pi/43));
B12=2*(sin(2*pi/43)+v2*sin(82*pi/43)+v4*sin(8*pi/43)+v6*sin(70*pi/43)+v1*sin(32*pi/43)+v3*sin(22*pi/43)+v5*sin(42*pi/43));
B22=2*(sin(12*pi/43)+v2*sin(62*pi/43)+v4*sin(48*pi/43)+v6*sin(76*pi/43)+v1*sin(20*pi/43)+v3*sin(46*pi/43)+v5*sin(80*pi/43));
B32=2*(sin(72*pi/43)+v2*sin(28*pi/43)+v4*sin(30*pi/43)+v6*sin(26*pi/43)+v1*sin(34*pi/43)+v3*sin(18*pi/43)+v5*sin(50*pi/43));
B13=2*(sin(2*pi/43)+v3*sin(82*pi/43)+v6*sin(8*pi/43)+v2*sin(70*pi/43)+v5*sin(32*pi/43)+v1*sin(22*pi/43)+v4*sin(42*pi/43));
B23=2*(sin(12*pi/43)+v3*sin(62*pi/43)+v6*sin(48*pi/43)+v2*sin(76*pi/43)+v5*sin(20*pi/43)+v1*sin(46*pi/43)+v4*sin(80*pi/43));
B33=2*(sin(72*pi/43)+v3*sin(28*pi/43)+v6*sin(30*pi/43)+v2*sin(26*pi/43)+v5*sin(34*pi/43)+v1*sin(18*pi/43)+v4*sin(50*pi/43));
B14=2*(sin(2*pi/43)+v4*sin(82*pi/43)+v1*sin(8*pi/43)+v5*sin(70*pi/43)+v2*sin(32*pi/43)+v6*sin(22*pi/43)+v3*sin(42*pi/43));
B24=2*(sin(12*pi/43)+v4*sin(62*pi/43)+v1*sin(48*pi/43)+v5*sin(76*pi/43)+v2*sin(20*pi/43)+v6*sin(46*pi/43)+v3*sin(80*pi/43));
B34=2*(sin(72*pi/43)+v4*sin(28*pi/43)+v1*sin(30*pi/43)+v5*sin(26*pi/43)+v2*sin(34*pi/43)+v6*sin(18*pi/43)+v3*sin(50*pi/43));
B15=2*(sin(2*pi/43)+v5*sin(82*pi/43)+v3*sin(8*pi/43)+v1*sin(70*pi/43)+v6*sin(32*pi/43)+v4*sin(22*pi/43)+v2*sin(42*pi/43));
B25=2*(sin(12*pi/43)+v5*sin(62*pi/43)+v3*sin(48*pi/43)+v1*sin(76*pi/43)+v6*sin(20*pi/43)+v4*sin(46*pi/43)+v2*sin(80*pi/43));
B35=2*(sin(72*pi/43)+v5*sin(28*pi/43)+v3*sin(30*pi/43)+v1*sin(26*pi/43)+v6*sin(34*pi/43)+v4*sin(18*pi/43)+v2*sin(50*pi/43));
B16=2*(sin(2*pi/43)+v6*sin(82*pi/43)+v5*sin(8*pi/43)+v4*sin(70*pi/43)+v3*sin(32*pi/43)+v2*sin(22*pi/43)+v1*sin(42*pi/43));
B26=2*(sin(12*pi/43)+v6*sin(62*pi/43)+v5*sin(48*pi/43)+v4*sin(76*pi/43)+v3*sin(20*pi/43)+v2*sin(46*pi/43)+v1*sin(80*pi/43));
B36=2*(sin(72*pi/43)+v6*sin(28*pi/43)+v5*sin(30*pi/43)+v4*sin(26*pi/43)+v3*sin(34*pi/43)+v2*sin(18*pi/43)+v1*sin(50*pi/43));
那么通過Magma小程序演算：
B10=(sqrt(43)+w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
B20=(sqrt(43)+w2*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+w1*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
B30=(sqrt(43)+(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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2.?14*sin(2*k*pi/43)的6個(gè)7次復(fù)根式項(xiàng)。設(shè)其內(nèi)表達(dá)式寫成Qyz=Yy1+Yy2*v1^z+Yy3*v3^z+Yy4*v4^z+Yy5*v5^z+Yy6*v6^z，并且還能消去v6^z項(xiàng)。
Q11=B11^7; Q12=B12^7; Q13=B13^7; Q14=B14^7; Q15=B15^7; Q16=B16^7; Q10=B10^7;
Q21=B21^7; Q22=B22^7; Q23=B23^7; Q24=B24^7; Q25=B25^7; Q26=B26^7; Q20=B20^7;
Q31=B31^7; Q32=B32^7; Q33=B33^7; Q34=B34^7; Q35=B35^7; Q36=B36^7; Q30=B30^7;
爾后依次求解中間量Yy0, Yy1......, Yy6。
I.?Y10=(Q11+Q12+Q13+Q14+Q15+Q16+Q10)/7;
Y20=(Q21+Q22+Q23+Q24+Q25+Q26+Q20)/7;
Y30=(Q31+Q32+Q33+Q34+Q35+Q36+Q30)/7;
從而Y10=(11600*sqrt(43)+(-8171.5+1641.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8171.5-1641.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y20=(11600*sqrt(43)+(-8171.5+1641.5*sqrt(-3))*w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8171.5-1641.5*sqrt(-3))*w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y30=(11600*sqrt(43)+(-8171.5+1641.5*sqrt(-3))*w2*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8171.5-1641.5*sqrt(-3))*w1*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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II.?Y11=(Q11/v1+Q12/v2+Q13/v3+Q14/v4+Q15/v5+Q16/v6+Q10)/7;
Y21=(Q21/v1+Q22/v2+Q23/v3+Q24/v4+Q25/v5+Q26/v6+Q20)/7;
Y31=(Q31/v1+Q32/v2+Q33/v3+Q34/v4+Q35/v5+Q36/v6+Q30)/7;
從而Y11=(6209*sqrt(43)+(-7514.5+2425.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7514.5-2425.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y21=(6209*sqrt(43)+(-7514.5+2425.5*sqrt(-3))*w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7514.5-2425.5*sqrt(-3))*w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y31=(6209*sqrt(43)+(-7514.5+2425.5*sqrt(-3))*w2*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7514.5-2425.5*sqrt(-3))*w1*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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III.?Y12=(Q11/v2+Q12/v4+Q13/v6+Q14/v1+Q15/v3+Q16/v5+Q10)/7;
Y22=(Q21/v2+Q22/v4+Q23/v6+Q24/v1+Q25/v3+Q26/v5+Q20)/7;
Y32=(Q31/v2+Q32/v4+Q33/v6+Q34/v1+Q35/v3+Q36/v5+Q30)/7;
從而Y12=(5838*sqrt(43)+(-3895.5+633.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-3895.5-633.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y22=(5838*sqrt(43)+(-3895.5+633.5*sqrt(-3))*w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-3895.5-633.5*sqrt(-3))*w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y32=(5838*sqrt(43)+(-3895.5+633.5*sqrt(-3))*w2*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-3895.5-633.5*sqrt(-3))*w1*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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IV.?Y13=(Q11/v3+Q12/v6+Q13/v2+Q14/v5+Q15/v1+Q16/v4+Q10)/7;
Y23=(Q21/v3+Q22/v6+Q23/v2+Q24/v5+Q25/v1+Q26/v4+Q20)/7;
Y33=(Q31/v3+Q32/v6+Q33/v2+Q34/v5+Q35/v1+Q36/v4+Q30)/7;
從而Y13=(1806*sqrt(43)+(-850.5+416.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-850.5-416.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y23=(1806*sqrt(43)+(-850.5+416.5*sqrt(-3))*w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-850.5-416.5*sqrt(-3))*w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y33=(1806*sqrt(43)+(-850.5+416.5*sqrt(-3))*w2*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-850.5-416.5*sqrt(-3))*w1*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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V.?Y14=(Q11/v4+Q12/v1+Q13/v5+Q14/v2+Q15/v6+Q16/v3+Q10)/7;
Y24=(Q21/v4+Q22/v1+Q23/v5+Q24/v2+Q25/v6+Q26/v3+Q20)/7;
Y34=(Q31/v4+Q32/v1+Q33/v5+Q34/v2+Q35/v6+Q36/v3+Q30)/7;
從而Y14=(-3108*sqrt(43)+(-367.5+738.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-367.5-738.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y24=(-3108*sqrt(43)+(-367.5+738.5*sqrt(-3))*w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-367.5-738.5*sqrt(-3))*w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y34=(-3108*sqrt(43)+(-367.5+738.5*sqrt(-3))*w2*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-367.5-738.5*sqrt(-3))*w1*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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VI.?Y15=(Q11/v5+Q12/v3+Q13/v1+Q14/v6+Q15/v4+Q16/v2+Q10)/7;
Y25=(Q21/v5+Q22/v3+Q23/v1+Q24/v6+Q25/v4+Q26/v2+Q20)/7;
Y35=(Q31/v5+Q32/v3+Q33/v1+Q34/v6+Q35/v4+Q36/v2+Q30)/7;
從而Y15=(5495*sqrt(43)+(-1382.5-920.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1382.5+920.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y25=(5495*sqrt(43)+(-1382.5-920.5*sqrt(-3))*w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1382.5+920.5*sqrt(-3))*w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y35=(5495*sqrt(43)+(-1382.5-920.5*sqrt(-3))*w2*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1382.5+920.5*sqrt(-3))*w1*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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VII.?Y16=(Q11/v6+Q12/v5+Q13/v4+Q14/v3+Q15/v2+Q16/v1+Q10)/7;
Y26=(Q21/v6+Q22/v5+Q23/v4+Q24/v3+Q25/v2+Q26/v1+Q20)/7;
Y36=(Q31/v6+Q32/v5+Q33/v4+Q34/v3+Q35/v2+Q36/v1+Q30)/7;
從而Y16=(4711*sqrt(43)+(-7122.5+3041.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7122.5-3041.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y26=(4711*sqrt(43)+(-7122.5+3041.5*sqrt(-3))*w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7122.5-3041.5*sqrt(-3))*w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
Y36=(4711*sqrt(43)+(-7122.5+3041.5*sqrt(-3))*w2*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7122.5-3041.5*sqrt(-3))*w1*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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借助1+v1+......+v6=0消除含Xy6的項(xiàng)，最終得出：
Q11=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5)/3;
Q12=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3)/3;
Q13=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1)/3;
Q14=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6)/3;
Q15=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4)/3;
Q16=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2)/3;
Q21=(6889*sqrt(43)+(2624.5+175.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(2624.5-175.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(1120+112*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1120-112*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(1127*sqrt(43)+(1998.5+2817.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1998.5-2817.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(-2905*sqrt(43)+(801.5+4448.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(801.5-4448.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(-7819*sqrt(43)+(77+4529*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(77-4529*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(784*sqrt(43)+(3073+4851*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3073-4851*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5)/3;
Q22=(6889*sqrt(43)+(2624.5+175.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(2624.5-175.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(1120+112*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1120-112*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(1127*sqrt(43)+(1998.5+2817.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1998.5-2817.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(-2905*sqrt(43)+(801.5+4448.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(801.5-4448.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(-7819*sqrt(43)+(77+4529*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(77-4529*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(784*sqrt(43)+(3073+4851*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3073-4851*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3)/3;
Q23=(6889*sqrt(43)+(2624.5+175.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(2624.5-175.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(1120+112*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1120-112*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(1127*sqrt(43)+(1998.5+2817.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1998.5-2817.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(-2905*sqrt(43)+(801.5+4448.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(801.5-4448.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(-7819*sqrt(43)+(77+4529*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(77-4529*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(784*sqrt(43)+(3073+4851*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3073-4851*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1)/3;
Q24=(6889*sqrt(43)+(2624.5+175.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(2624.5-175.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(1120+112*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1120-112*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(1127*sqrt(43)+(1998.5+2817.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1998.5-2817.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(-2905*sqrt(43)+(801.5+4448.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(801.5-4448.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(-7819*sqrt(43)+(77+4529*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(77-4529*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(784*sqrt(43)+(3073+4851*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3073-4851*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6)/3;
Q25=(6889*sqrt(43)+(2624.5+175.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(2624.5-175.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(1120+112*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1120-112*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(1127*sqrt(43)+(1998.5+2817.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1998.5-2817.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(-2905*sqrt(43)+(801.5+4448.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(801.5-4448.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(-7819*sqrt(43)+(77+4529*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(77-4529*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(784*sqrt(43)+(3073+4851*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3073-4851*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4)/3;
Q26=(6889*sqrt(43)+(2624.5+175.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(2624.5-175.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(1120+112*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1120-112*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(1127*sqrt(43)+(1998.5+2817.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(1998.5-2817.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(-2905*sqrt(43)+(801.5+4448.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(801.5-4448.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(-7819*sqrt(43)+(77+4529*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(77-4529*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(784*sqrt(43)+(3073+4851*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3073-4851*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2)/3;
Q31=(6889*sqrt(43)+(-1575.5+1224.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1575.5-1224.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-728+504*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-728-504*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(1127*sqrt(43)+(-5225.5-409.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-5225.5+409.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(-2905*sqrt(43)+(-7073.5-1823.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7073.5+1823.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(-7819*sqrt(43)+(-6832-2226*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-6832+2226*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(784*sqrt(43)+(-8813-889*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8813+889*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5)/3;
Q32=(6889*sqrt(43)+(-1575.5+1224.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1575.5-1224.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-728+504*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-728-504*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(1127*sqrt(43)+(-5225.5-409.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-5225.5+409.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(-2905*sqrt(43)+(-7073.5-1823.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7073.5+1823.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(-7819*sqrt(43)+(-6832-2226*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-6832+2226*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(784*sqrt(43)+(-8813-889*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8813+889*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3)/3;
Q33=(6889*sqrt(43)+(-1575.5+1224.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1575.5-1224.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-728+504*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-728-504*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(1127*sqrt(43)+(-5225.5-409.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-5225.5+409.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(-2905*sqrt(43)+(-7073.5-1823.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7073.5+1823.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(-7819*sqrt(43)+(-6832-2226*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-6832+2226*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(784*sqrt(43)+(-8813-889*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8813+889*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1)/3;
Q34=(6889*sqrt(43)+(-1575.5+1224.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1575.5-1224.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-728+504*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-728-504*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(1127*sqrt(43)+(-5225.5-409.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-5225.5+409.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(-2905*sqrt(43)+(-7073.5-1823.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7073.5+1823.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(-7819*sqrt(43)+(-6832-2226*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-6832+2226*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(784*sqrt(43)+(-8813-889*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8813+889*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6)/3;
Q35=(6889*sqrt(43)+(-1575.5+1224.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1575.5-1224.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-728+504*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-728-504*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(1127*sqrt(43)+(-5225.5-409.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-5225.5+409.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(-2905*sqrt(43)+(-7073.5-1823.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7073.5+1823.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(-7819*sqrt(43)+(-6832-2226*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-6832+2226*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(784*sqrt(43)+(-8813-889*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8813+889*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4)/3;
Q36=(6889*sqrt(43)+(-1575.5+1224.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1575.5-1224.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-728+504*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-728-504*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(1127*sqrt(43)+(-5225.5-409.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-5225.5+409.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(-2905*sqrt(43)+(-7073.5-1823.5*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-7073.5+1823.5*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(-7819*sqrt(43)+(-6832-2226*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-6832+2226*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(784*sqrt(43)+(-8813-889*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-8813+889*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2)/3;
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3.?以By0, Qy1^(1/7), ......Qy6^(1/7)線性表示sin(2*k*pi/43)，用矩陣方程表達(dá)顯然更加簡潔。必然用到7次單位根系數(shù)，每7個(gè)以-2為底數(shù)的k值對(duì)應(yīng)同一個(gè)B、Q值，而每個(gè)k值唯一對(duì)應(yīng)6個(gè)7次單位根系數(shù)的一套組合。列舉全部的sin(2*k*pi/43)值如下：
I.以k=1為開頭
sin(2*pi/43)=(B10+v5*Q11^(1/7)+v2*Q16^(1/7)+v3*Q12^(1/7)+v4*Q15^(1/7)+v6*Q13^(1/7)+v1*Q14^(1/7))/14;
sin(82*pi/43)=(B10+v4*Q11^(1/7)+v3*Q16^(1/7)+v1*Q12^(1/7)+v6*Q15^(1/7)+v3*Q13^(1/7)+v4*Q14^(1/7))/14;
sin(8*pi/43)=(B10+v3*Q11^(1/7)+v4*Q16^(1/7)+v6*Q12^(1/7)+v1*Q15^(1/7)+Q13^(1/7)+Q14^(1/7))/14;
sin(70*pi/43)=(B10+v2*Q11^(1/7)+v5*Q16^(1/7)+v4*Q12^(1/7)+v3*Q15^(1/7)+v4*Q13^(1/7)+v3*Q14^(1/7))/14;
sin(32*pi/43)=(B10+v1*Q11^(1/7)+v6*Q16^(1/7)+v2*Q12^(1/7)+v5*Q15^(1/7)+v1*Q13^(1/7)+v6*Q14^(1/7))/14;
sin(22*pi/43)=(B10+Q11^(1/7)+Q16^(1/7)+Q12^(1/7)+Q15^(1/7)+v5*Q13^(1/7)+v2*Q14^(1/7))/14;
sin(42*pi/43)=(B10+v6*Q11^(1/7)+v1*Q16^(1/7)+v5*Q12^(1/7)+v2*Q15^(1/7)+v2*Q13^(1/7)+v5*Q14^(1/7))/14;
II.以k=6為開頭
sin(12*pi/43)=(B20+v5*Q21^(1/7)+v2*Q26^(1/7)+Q22^(1/7)+Q25^(1/7)+v6*Q23^(1/7)+v1*Q24^(1/7))/14;
sin(62*pi/43)=(B20+v4*Q21^(1/7)+v3*Q26^(1/7)+v5*Q22^(1/7)+v2*Q25^(1/7)+v3*Q23^(1/7)+v4*Q24^(1/7))/14;
sin(48*pi/43)=(B20+v3*Q21^(1/7)+v4*Q26^(1/7)+v3*Q22^(1/7)+v4*Q25^(1/7)+Q23^(1/7)+Q24^(1/7))/14;
sin(76*pi/43)=(B20+v2*Q21^(1/7)+v5*Q26^(1/7)+v1*Q22^(1/7)+v6*Q25^(1/7)+v4*Q23^(1/7)+v3*Q24^(1/7))/14;
sin(20*pi/43)=(B20+v1*Q21^(1/7)+v6*Q26^(1/7)+v6*Q22^(1/7)+v1*Q25^(1/7)+v1*Q23^(1/7)+v6*Q24^(1/7))/14;
sin(46*pi/43)=(B20+Q21^(1/7)+Q26^(1/7)+v4*Q22^(1/7)+v3*Q25^(1/7)+v5*Q23^(1/7)+v2*Q24^(1/7))/14;
sin(80*pi/43)=(B20+v6*Q21^(1/7)+v1*Q26^(1/7)+v2*Q22^(1/7)+v5*Q25^(1/7)+v2*Q23^(1/7)+v5*Q24^(1/7))/14;
III.以k=36為開頭
sin(72*pi/43)=(B30+v3*Q31^(1/7)+v4*Q36^(1/7)+v3*Q32^(1/7)+v4*Q35^(1/7)+v2*Q33^(1/7)+v5*Q34^(1/7))/14;
sin(28*pi/43)=(B30+v2*Q31^(1/7)+v5*Q36^(1/7)+v1*Q32^(1/7)+v6*Q35^(1/7)+v6*Q33^(1/7)+v1*Q34^(1/7))/14;
sin(30*pi/43)=(B30+v1*Q31^(1/7)+v6*Q36^(1/7)+v6*Q32^(1/7)+v1*Q35^(1/7)+v3*Q33^(1/7)+v4*Q34^(1/7))/14;
sin(26*pi/43)=(B30+Q31^(1/7)+Q36^(1/7)+v4*Q32^(1/7)+v3*Q35^(1/7)+Q33^(1/7)+Q34^(1/7))/14;
sin(34*pi/43)=(B30+v6*Q31^(1/7)+v1*Q36^(1/7)+v2*Q32^(1/7)+v5*Q35^(1/7)+v4*Q33^(1/7)+v3*Q34^(1/7))/14;
sin(18*pi/43)=(B30+v5*Q31^(1/7)+v2*Q36^(1/7)+Q32^(1/7)+Q35^(1/7)+v1*Q33^(1/7)+v6*Q34^(1/7))/14;
sin(50*pi/43)=(B30+v4*Q31^(1/7)+v3*Q36^(1/7)+v5*Q32^(1/7)+v2*Q35^(1/7)+v5*Q33^(1/7)+v2*Q34^(1/7))/14;
格式很優(yōu)美。讀者可以推斷出v的取值規(guī)律。
四．43次單位根舉例
根據(jù)Euler公式exp(j*x)=cos(x)+j*sin(x)得到輻角為2*pi/43的解析解是：
exp(2*pi*j/43)=(A10+v1*P11^(1/7)+v6*P16^(1/7)+v2*P12^(1/7)+v5*P15^(1/7)+v1*P13^(1/7)+v6*P14^(1/7)+j*(B10+v5*Q11^(1/7)+v2*Q16^(1/7)+v3*Q12^(1/7)+v4*Q15^(1/7)+v6*Q13^(1/7)+v1*Q14^(1/7)))/14.
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其中，在上一章聲明了，vi是輻角2*i*pi/7的單位根。其他參數(shù)：
I.?
A10=(-1+(-172-129*sqrt(-3))^(1/3)+(-172+129*sqrt(-3))^(1/3))/3;
B10=(sqrt(43)+w1*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+w2*(-11*sqrt(43)+24*sqrt(-129))^(1/3))/3;
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II.?
(1)
P11=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5)/3;
P12=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3)/3;
P13=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1)/3;
P14=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6)/3;
P15=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v1+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4)/3;
P16=(-42785+(2568.5+7342.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(2568.5-7342.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3)+(207690+(16485-3381*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(16485+3381*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v6+(263977+(22221.5+1669.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(22221.5-1669.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v5+(72541+(17328.5+15284.5*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(17328.5-15284.5*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v4+(124915+(12110+5138*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(12110-5138*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v3+(156520+(9590-11438*sqrt(-3))*(-172-129*sqrt(-3))^(1/3)+(9590+11438*sqrt(-3))*(-172+129*sqrt(-3))^(1/3))*v2)/3;
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(2)
Q11=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5)/3;
Q12=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3)/3;
Q13=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1)/3;
Q14=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6)/3;
Q15=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v1+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4)/3;
Q16=(6889*sqrt(43)+(-1049-1400*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-1049+1400*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3)+(1498*sqrt(43)+(-392-616*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(-392+616*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v6+(1127*sqrt(43)+(3227-2408*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(3227+2408*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v5+(-2905*sqrt(43)+(6272-2625*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6272+2625*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v4+(-7819*sqrt(43)+(6755-2303*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(6755+2303*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v3+(784*sqrt(43)+(5740-3962*sqrt(-3))*(-11*sqrt(43)-24*sqrt(-129))^(1/3)+(5740+3962*sqrt(-3))*(-11*sqrt(43)+24*sqrt(-129))^(1/3))*v2)/3.
(全章終)
