Families of Polynomials of degree n with the dihedral group D_n as Galois group: (n, u, v, b, c, d, f are elements of the rationals) n = 3 F := x^3-n*x^2+u*v*x+v^2; n = 4 F := x^4-n*x^3+(-b+n*b)*x^2+2*b^2*x-b^3; n = 5 F := x^5-n*x^4-(-c^3-2*n*c+c^2+c)*x^3-(c^3+n*c^2-3*c^2)*x^2-(-c^4+3*c^3)*x+c^4; n = 6 F := x^6-n*x^5+(-c-4*c^3-5*c^2+3*n*c+2*n*c^2)*x^4+(8*c^4+10*c^3-4*n*c^3-3*n*c^2+4*c^2-n*c^4+2*c^5)*x^3+ (-6*c^3-2*c^6+2*n*c^4+n*c^3+n*c^5-10*c^5-14*c^4)*x^2+(4*c^4+12*c^5+12*c^6+4*c^7)*x-c^5-4*c^6-c^9-6*c^7-4*c^8; with substitution x = c*(c+1)/x F transforms to F := x^6-4*x^5+(2*c-n+6)*x^4+(-2*c^2-6*c+n*c-4+3*n)*x^3+(4*c^2-2*n*c+5*c+1-3*n)*x^2+n*(c+1)*x-c*(c+1)^2; n = 7 F := x^7-n*x^6-(2*n*d+3*d^6-4*d^2+12*d^3-16*d^4+5*d^5-d^7-2*n*d^3+d)*x^5- (4*d^9+50*d^7+23*d^5+d^3+2*n*d^5-25*d^8-6*n*d^4-7*d^4+n*d^2+n*d^6+2*n*d^3-46*d^6)*x^4- (14*d^9-4*n*d^5-12*d^10+2*n*d^4+6*d^11-12*d^8-2*n*d^8-d^12+4*n*d^7+2*d^6-d^4+4*d^7)*x^3- (d^13+4*d^8-15*d^9-6*d^12-3*d^6+5*d^10+6*n*d^8+8*d^11+n*d^6+n*d^10-4*n*d^9+6*d^7-4*n*d^7)*x^2- (-d^15-5*d^11+8*d^14+25*d^12+12*d^9-14*d^10-22*d^13-3*d^8)*x+15*d^12+d^16+d^10-20*d^13+15*d^14-6*d^11-6*d^15; with substitution x = d^2*(d-1)/x F transforms to F := x^7+(-3+d^2-3*d)*x^6+(-d^3+6*d-n+2*d^2+3)*x^5+(2*n-1-4*d^2-3*d^4+d^5-3*d+2*n*d)*x^4+ (-4*n*d+5*d^2-12*d^3-4*d^5-n-n*d^2-d+17*d^4)*x^3+ (-3*d^2+9*d^3+d^6+2*n*d+2*n*d^2-7*d^4+d-2*d^5)*x^2-n*x*d^2+d^4*(d-1); n = 8 F := x^8-n*x^7+(d-1)*d*(-40*d^5+68*d^4-28*d^3-6*d^2+5*d^2*n+6*d+2*d*n-1-2*n)*x^6- d^2*(2*d-1)*(d-1)^2*(-32*d^6-24*d^5+124*d^4+4*d^3*n-104*d^3+9*d^2*n+34*d^2-4*d-2*d*n-n)*x^5+ d^4*(2*d-1)^2*(d-1)^3*(-8*d^6-52*d^5+68*d^4-17*d^3+d^3*n-12*d^2+8*d^2*n+7*d+3*d*n-1-2*n)*x^4- d^6*(2*d-1)^3*(d-1)^4*(-16*d^5-8*d^4+4*d^3+2*d^2*n+4*d*n-n)*x^3+ d^9*(2*d-1)^4*(d-1)^5*(-8*d^3-14*d^2+8*d-1+n)*x^2+2*d^12*(-1+4*d)*(2*d-1)^5*(d-1)^6*x-d^15*(2*d-1)^6*(d-1)^7; with substitution x=d^3*(2*d-1)*(d-1)/x F transforms to F := x^8+(2-8*d)*x^7+(-n+8*d^3+14*d^2-8*d+1)*x^6+(2*n*d^2-8*d^4-n-16*d^5+4*d^3+4*n*d)*x^5+ (-68*d^5-3*n*d^2+17*d^4+8*d^7+2*n*d+52*d^6+12*d^3-7*d^2-8*n*d^3+d-n*d^4)*x^4+ d^2*(-32*d^6-24*d^5+124*d^4+(4*n-104)*d^3+(9*n+34)*d^2+(-2*n-4)*d-n)*x^3+ d^4*(40*d^5-68*d^4+28*d^3+(-5*n+6)*d^2+(-2*n-6)*d+2*n+1)*x^2+(d^6*n*(2*d-1))*x-d^9*(d-1)*(2*d-1)^2; n = 9 F := x^9-n*x^8 + f*(f-1)*(f^9-3*f^8-3*f^7+15*f^6-39*f^5+54*f^4-48*f^3+2*n*f^3+27*f^2+2*n*f-6*f+2*n+1)*x^7 - f^2*(f-1)^2*(4*f^11-24*f^10+69*f^9-174*f^8+303*f^7-399*f^6+n*f^6+369*f^5+2*n*f^5-237*f^4+99*f^3+8*n*f^3-20*f^2-3*n*f^2+3*f+6*n*f+n)*x^6+ f^4*(f^2-f+1)*(f-1)^3*(f^11-6*f^10+29*f^9-111*f^8+225*f^7-336*f^6+327*f^5+2*n*f^5-225*f^4+93*f^3+8*n*f^3-20*f^2+3*f+6*n*f-1+4*n)*x^5- f^6*(f^2-f+1)^2*(f-1)^4*(4*f^9-21*f^8+45*f^7-83*f^6+83*f^5+n*f^4-71*f^4+2*n*f^3+25*f^3-13*f^2+4*n*f^2+2*n*f+f-5+6*n)*x^4+ f^8*(f^2-f+1)^3*(f-1)^5*(f^8-4*f^7+5*f^6-7*f^5-4*f^4-2*f^3-14*f^2+2*n*f^2-10+4*n)*x^3- f^10*(f^2-f+1)^4*(f-1)^6*(f^5-3*f^4+2*f^3-11*f^2-10+n)*x^2+f^12*(f^3-3*f^2-5)*(f^2-f+1)^5*(f-1)^7*x+f^14*(f^2-f+1)^6*(f-1)^8; with substitution x=f^2*(f-1)*(f^2-f+1)/x F transforms to F := x^9 + (f^3-3*f^2-5)*x^8 + (-f^5+3*f^4-2*f^3+11*f^2-n+10)*x^7 + (f^8-4*f^7+5*f^6-7*f^5-4*f^4-2*f^3+(-14+2*n)*f^2+4*n-10)*x^6 + (-4*f^9+21*f^8-45*f^7+83*f^6-83*f^5+(71-n)*f^4+(-25-2*n)*f^3+(-4*n+13)*f^2+(-1-2*n)*f+5-6*n)*x^5 + (f^11-6*f^10+29*f^9-111*f^8+225*f^7-336*f^6+(2*n+327)*f^5-225*f^4+(93+8*n)*f^3-20*f^2+(3+6*n)*f+4*n-1)*x^4 + (-4*f^11+24*f^10-69*f^9+174*f^8-303*f^7+(399-n)*f^6-(369+2*n)*f^5+237*f^4-(99+8*n)*f^3+(3*n+20)*f^2-(6*n+3)*f-n)*x^3 + f*(f^2-f+1)*(f^9-3*f^8-3*f^7+15*f^6-39*f^5+54*f^4+(2*n-48)*f^3+27*f^2+(2*n-6)*f+1+2*n)*x^2 + (-f^2*n*(f^2-f+1)^2)*x + f^4*(f-1)*(f^2-f+1)^3; n = 10 F := x^10-n*x^9 + f*(f-1)*(74*f^8-365*f^7+612*f^6-534*f^5-n*f^4+335*f^4-176*f^3+3*n*f^3+15*n*f^2+65*f^2-12*n*f-13*f+1+2*n)*x^8 + f^2*(2*f-1)*(f-1)^2*(64*f^11-480*f^10+1144*f^9-852*f^8-160*f^7+538*f^6-436*f^5+12*n*f^5+308*f^4-40*n*f^4-20*n*f^3-154*f^3+38*n*f^2+40*f^2-4*f-12*n*f+n)*x^7+ f^4*(2*f-1)^2*(f-1)^3*(16*f^13-160*f^12+448*f^11+110*f^10-2199*f^9+3121*f^8-1752*f^7+2*n*f^7+406*f^6-12*n*f^6+111*f^5-20*n*f^5-281*f^4+138*n*f^4+222*f^3-67*n*f^3-84*f^2-19*n*f^2+15*f+15*n*f-1-2*n)*x^6+ (-f^6*(2*f-1)^3*(f-1)^4*(48*f^13-480*f^12+1692*f^11-2274*f^10+12*f^9+2918*f^8-3722*f^7+12*n*f^7+2918*f^6-75*n*f^6-1746*f^5+98*n*f^5+758*f^4+87*n*f^4-210*f^3-118*n*f^3+32*f^2+30*n*f^2-2*f+2*n*f-n))*x^5+ f^9*(-3*f+f^2+1)*(2*f-1)^4*(f-1)^5*(54*f^10-379*f^9+935*f^8-1144*f^7+1060*f^6-n*f^6-619*f^5+6*n*f^5+205*f^4+7*n*f^4-64*f^3-54*n*f^3+30*f^2+14*n*f^2-9*f+10*n*f+1-3*n)*x^4+ f^12*(-3*f+f^2+1)^2*(2*f-1)^5*(f-1)^6*(4*f^7-62*f^6+120*f^5+68*f^4-70*f^3+2*n*f^3-24*f^2-6*n*f^2+24*f-6*n*f-4+3*n)*x^3+ f^15*(-3*f+f^2+1)^3*(2*f-1)^6*(f-1)^7*(4*f^6-20*f^5+4*f^4+58*f^3-24*f+6-n)*x^2+ (-4*f^18*(f+1)*(-3*f+f^2+1)^5*(2*f-1)^7*(f-1)^8)*x + f^21*(-3*f+f^2+1)^5*(2*f-1)^8*(f-1)^9; with substitution x=f^3*(f-1)*(f^2-3*f+1)*(2*f-1)/x F transforms to F:= x^10 + (-4*(f+1)*(-3*f+f^2+1))*x^9 + (4*f^6-20*f^5+4*f^4+58*f^3-24*f+6-n)*x^8 + (4*f^7-62*f^6+120*f^5+68*f^4-70*f^3+2*n*f^3-24*f^2-6*n*f^2+24*f-6*n*f-4+3*n)*x^7 + (54*f^10-379*f^9+935*f^8-1144*f^7+1060*f^6-n*f^6-619*f^5+6*n*f^5+205*f^4+7*n*f^4-64*f^3-54*n*f^3+30*f^2+14*n*f^2-9*f+10*n*f+1-3*n)*x^6 + (210*f^3-32*f^2-48*f^13-98*n*f^5-2*n*f+75*n*f^6+2*f+118*n*f^3-2918*f^8-12*f^9-1692*f^11+n-758*f^4+480*f^12-12*n*f^7-2918*f^6-30*n*f^2-87*n*f^4+3722*f^7+1746*f^5+2274*f^10)*x^5 + (f*(-3*f+f^2+1)*(16*f^13-160*f^12+448*f^11+110*f^10-2199*f^9+3121*f^8-1752*f^7+2*n*f^7+406*f^6-12*n*f^6+111*f^5-20*n*f^5-281*f^4+138*n*f^4+222*f^3-67*n*f^3-84*f^2-19*n*f^2+15*f+15*n*f-1-2*n))*x^4 + (f^2*(-3*f+f^2+1)^2*(64*f^11-480*f^10+1144*f^9-852*f^8-160*f^7+538*f^6-436*f^5+12*n*f^5+308*f^4-40*n*f^4-20*n*f^3-154*f^3+38*n*f^2+40*f^2-4*f-12*n*f+n))*x^3 + (f^4*(-3*f+f^2+1)^3*(74*f^8-365*f^7+612*f^6-534*f^5-n*f^4+335*f^4-176*f^3+3*n*f^3+15*n*f^2+65*f^2-12*n*f-13*f+1+2*n))*x^2 + (-n*f^6*(2*f-1)*(-3*f+f^2+1)^4)*x + (f^9*(f-1)*(2*f-1)^2*(-3*f+f^2+1)^5);