Q(A_n) is the extension of Q joining all sqrt((Product_{i=1..k} lambda_i)*), where lambda_1,...,lambda_k are distinct odd numbers sum to n, and m* = (-1)^((m-1)/2)*m. 3 = 3: Q(A_3) = Q(sqrt(-3)) 4 = 1+3: Q(A_4) = Q(sqrt(-3)) 5 = 5: Q(A_5) = Q(sqrt(5)) 6 = 1+5: Q(A_6) = Q(sqrt(5)) 7 = 7: Q(A_7) = Q(sqrt(-7)) 8 = 1+7 = 3+5: Q(A_8) = Q(sqrt(-3*5),sqrt(-7)) 9 = 9 = 1+3+5: Q(A_9) = Q(sqrt(-3*5)) 10 = 3+7 = 1+9: Q(A_10) = Q(sqrt(3*7)) 11 = 11 = 1+3+7: Q(A_11) = Q(sqrt(3*7),sqrt(-11)) 12 = 1+11 = 3+9 = 5+7: Q(A_12) = Q(sqrt(-3),sqrt(-5*7),sqrt(-11)) 13 = 13 = 1+3+9 = 1+5+7: Q(A_13) = Q(sqrt(-3),sqrt(-5*7),sqrt(13)) 14 = 1+13 = 3+11 = 5+9: Q(A_14) = Q(sqrt(3*11),sqrt(5),sqrt(13)) 15 = 15 = 1+3+11 = 1+5+9 = 3+5+7: Q(A_15) = Q(sqrt(-3),sqrt(5),sqrt(-7),sqrt(-11)) 16 = 1+15 = 3+13 = 5+11 = 7+9 = 1+3+5+7: Q(A_16) = Q(sqrt(-3*5),sqrt(3*11),sqrt(-3*13),sqrt(-7)) 17 = 17 = 1+3+13 = 1+5+11 = 1+7+9 = 3+5+9: Q(A_17) = Q(sqrt(-3*5),sqrt(3*11),sqrt(-3*13),sqrt(-7),sqrt(17)) 18 = 1+17 = 3+15 = 5+13 = 7+11 = 1+3+5+9 = Q(A_18) = Q(sqrt(-3),sqrt(5),sqrt(7*11),sqrt(13),sqrt(17)) 19 = 19 = 1+3+15 = 1+5+13 = 1+7+11 = 3+5+11 = 3+7+9: Q(A_19) = Q(sqrt(3*7),sqrt(3*11),sqrt(5),sqrt(13),sqrt(-19)) 20 = 1+19 = 3+17 = 5+15 = 7+13 = 9+11 = 1+3+5+11 = 1+3+7+9: Q(A_20) = Q(sqrt(-3),sqrt(5),sqrt(-7),sqrt(-11),sqrt(13),sqrt(17),sqrt(-19)) 21 = 21 = 1+3+17 = 1+5+15 = 1+7+13 = 1+9+11 = 3+5+13 = 3+7+11 = 5+7+9: Q(A_21) = Q(sqrt(-3),sqrt(5),sqrt(-7),sqrt(-11),sqrt(13),sqrt(17)) 22 = 1+21 = 3+19 = 5+17 = 7+15 = 9+13 = 1+3+5+13 = 1+3+7+11 = 1+5+7+9: Q(A_22) = Q(sqrt(-3),sqrt(5),sqrt(-7),sqrt(-11),sqrt(13),sqrt(17),sqrt(-19)) 23 = 23 = 1+3+19 = 1+5+17 = 1+7+15 = 1+9+13 = 3+5+15 = 3+7+13 = 3+9+11 = 5+7+11: Q(A_23) = Q(sqrt(3*7),sqrt(3*11),sqrt(3*19),sqrt(5),sqrt(13),sqrt(17),sqrt(-23)) 24 = 1+23 = 3+21 = 5+19 = 7+17 = 9+15 = 11+13 = 1+3+5+15 = 1+3+7+13 = 1+3+9+11 = 1+5+7+11 = 3+5+7+9: Q(A_24) = Q(sqrt(3*7),sqrt(3*11),sqrt(3*19),sqrt(5),sqrt(13),sqrt(17),sqrt(-23))