# fields

## 128 days ago by voloch

F.<a>=NumberField(x^3+x+1) print (a^2 + a - 1)*(a^2 + 1) print 1/(a^2+1) print 1/(a^2+a)
 -a^2 - a - 2 -a -2*a^2 + a - 3 -a^2 - a - 2 -a -2*a^2 + a - 3
K.<a>=NumberField(x^4 - 15*x^2 - 17*x + 1) L.<a1> = K.galois_closure(); L G = L.optimized_subfields(name='b');[z[0] for z in G]
 [Number Field in b0 with defining polynomial x - 1, Number Field in b1 with defining polynomial x^3 - 2*x^2 - 25*x + 67, Number Field in b2 with defining polynomial x^4 - 4*x^3 - 9*x^2 + 9*x + 4, Number Field in b3 with defining polynomial x^4 - 4*x^3 - 9*x^2 + 43*x - 30, Number Field in b4 with defining polynomial x^4 - 15*x^2 - 17*x + 1, Number Field in b5 with defining polynomial x^6 - 8*x^5 + 6*x^4 + 95*x^3 - 294*x^2 + 305*x - 97, Number Field in b6 with defining polynomial x^6 - 8*x^5 + 6*x^4 + 95*x^3 - 294*x^2 + 305*x - 97, Number Field in b7 with defining polynomial x^6 - 8*x^5 + 6*x^4 + 95*x^3 - 294*x^2 + 305*x - 97, Number Field in b8 with defining polynomial x^6 - 8*x^5 + 6*x^4 + 95*x^3 - 294*x^2 + 305*x - 97, Number Field in b9 with defining polynomial x^6 - 5*x^5 - 40*x^4 + 373*x^3 - 1076*x^2 + 1276*x - 488, Number Field in b10 with defining polynomial x^6 - 5*x^5 - 37*x^4 + 47*x^3 + 445*x^2 + 607*x + 179, Number Field in b11 with defining polynomial x^6 - 4*x^5 - 14*x^4 + 17*x^3 + 20*x^2 - 13*x + 1, Number Field in b12 with defining polynomial x^6 - 4*x^5 - 14*x^4 + 17*x^3 + 20*x^2 - 13*x + 1, Number Field in b13 with defining polynomial x^6 - 4*x^5 - 14*x^4 + 17*x^3 + 20*x^2 - 13*x + 1, Number Field in b14 with defining polynomial x^6 - 4*x^5 - 14*x^4 + 17*x^3 + 20*x^2 - 13*x + 1, Number Field in b15 with defining polynomial x^6 - 2*x^5 - 19*x^4 + 59*x^3 - 38*x^2 - 8*x + 8, Number Field in b16 with defining polynomial x^6 - 2*x^5 - 19*x^4 + 59*x^3 - 38*x^2 - 8*x + 8, Number Field in b17 with defining polynomial x^6 - 2*x^5 - 19*x^4 + 59*x^3 - 38*x^2 - 8*x + 8, Number Field in b18 with defining polynomial x^6 - 2*x^5 - 19*x^4 + 59*x^3 - 38*x^2 - 8*x + 8, Number Field in b19 with defining polynomial x^6 - x^5 - 50*x^4 - 183*x^3 - 232*x^2 - 64*x + 41, Number Field in b20 with defining polynomial x^6 - x^5 - 47*x^4 + 131*x^3 + 329*x^2 - 1471*x + 1237, Number Field in b21 with defining polynomial x^6 - 30*x^4 + 221*x^2 - 289, Number Field in b22 with defining polynomial x^12 - 4*x^11 - 49*x^10 + 165*x^9 + 832*x^8 - 2010*x^7 - 6062*x^6 + 7723*x^5 + 17567*x^4 - 6827*x^3 - 12504*x^2 + 1393*x + 2087] [Number Field in b0 with defining polynomial x - 1, Number Field in b1 with defining polynomial x^3 - 2*x^2 - 25*x + 67, Number Field in b2 with defining polynomial x^4 - 4*x^3 - 9*x^2 + 9*x + 4, Number Field in b3 with defining polynomial x^4 - 4*x^3 - 9*x^2 + 43*x - 30, Number Field in b4 with defining polynomial x^4 - 15*x^2 - 17*x + 1, Number Field in b5 with defining polynomial x^6 - 8*x^5 + 6*x^4 + 95*x^3 - 294*x^2 + 305*x - 97, Number Field in b6 with defining polynomial x^6 - 8*x^5 + 6*x^4 + 95*x^3 - 294*x^2 + 305*x - 97, Number Field in b7 with defining polynomial x^6 - 8*x^5 + 6*x^4 + 95*x^3 - 294*x^2 + 305*x - 97, Number Field in b8 with defining polynomial x^6 - 8*x^5 + 6*x^4 + 95*x^3 - 294*x^2 + 305*x - 97, Number Field in b9 with defining polynomial x^6 - 5*x^5 - 40*x^4 + 373*x^3 - 1076*x^2 + 1276*x - 488, Number Field in b10 with defining polynomial x^6 - 5*x^5 - 37*x^4 + 47*x^3 + 445*x^2 + 607*x + 179, Number Field in b11 with defining polynomial x^6 - 4*x^5 - 14*x^4 + 17*x^3 + 20*x^2 - 13*x + 1, Number Field in b12 with defining polynomial x^6 - 4*x^5 - 14*x^4 + 17*x^3 + 20*x^2 - 13*x + 1, Number Field in b13 with defining polynomial x^6 - 4*x^5 - 14*x^4 + 17*x^3 + 20*x^2 - 13*x + 1, Number Field in b14 with defining polynomial x^6 - 4*x^5 - 14*x^4 + 17*x^3 + 20*x^2 - 13*x + 1, Number Field in b15 with defining polynomial x^6 - 2*x^5 - 19*x^4 + 59*x^3 - 38*x^2 - 8*x + 8, Number Field in b16 with defining polynomial x^6 - 2*x^5 - 19*x^4 + 59*x^3 - 38*x^2 - 8*x + 8, Number Field in b17 with defining polynomial x^6 - 2*x^5 - 19*x^4 + 59*x^3 - 38*x^2 - 8*x + 8, Number Field in b18 with defining polynomial x^6 - 2*x^5 - 19*x^4 + 59*x^3 - 38*x^2 - 8*x + 8, Number Field in b19 with defining polynomial x^6 - x^5 - 50*x^4 - 183*x^3 - 232*x^2 - 64*x + 41, Number Field in b20 with defining polynomial x^6 - x^5 - 47*x^4 + 131*x^3 + 329*x^2 - 1471*x + 1237, Number Field in b21 with defining polynomial x^6 - 30*x^4 + 221*x^2 - 289, Number Field in b22 with defining polynomial x^12 - 4*x^11 - 49*x^10 + 165*x^9 + 832*x^8 - 2010*x^7 - 6062*x^6 + 7723*x^5 + 17567*x^4 - 6827*x^3 - 12504*x^2 + 1393*x + 2087]
K.<a>=NumberField(x^6-x+1) G = K.optimized_subfields(name='b');[z[0] for z in G]
 [Number Field in b0 with defining polynomial x - 1, Number Field in b1 with defining polynomial x^6 - x + 1] [Number Field in b0 with defining polynomial x - 1, Number Field in b1 with defining polynomial x^6 - x + 1]