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| On these pages, we present most recent results of our joint research, directly from the lab. |
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Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237 A. Derhem, Capitulation dans les extensions quadratiques de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Quebec, 1988 D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58 |
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Web master's e-mail address:
contact@algebra.at |
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| Sextic normal fields with cyclic or dihedral Galois group (2002/02/22) |
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Dan (02/02/14):
For my investigations into sextic fields I had to extend the
formula for the p-ray class rank from rational conductors to quadratic ideal conductors: let f be an integral ideal of a quadratic field K with discriminant d, then the p-rank ot the ray class group mod f of K is rho'(f,p) = rho(p) + rho''(f,p) - delta'_p(f) where rho''(f,p) = dim_{F_p}( K(f) / S_f*K(f)^p ) denotes the p-rank of the ray generator class group mod f, which is given by rho''(f,p) = t'' + w'' with t'' = #{ q prime ideal of K | v_q(f)>=1, Norm_{K|Q}(q)=+1(mod p) } and w'' concerns prime ideals q over p, with Norm_{K|Q}(q)=0(mod p): w'' = 0, if v_q(f)<=1 w'' = inertia degree of q, if v_q(f)>=2, q unramified w'' = 1, if v_q(f)=2, q ramified w'' = 2, if v_q(f)=3, q ramified, p=3, d=-3(mod 9) or v_q(f)>=3, q ramified, p>3 or p=3, d=+3(mod 9) w'' = 3, if v_q(f)>=4, q ramified, p=3, d=-3(mod 9) The implications of this generalized formula are in full accordance a) with the special case of a rational integer conductor f, b) with the PARI results on sextics in your last communication of 2002/02/12. I will further report on sextics in a forthcoming communication. I conjecture that we must distinguish multiplicities of discriminants from multiplicities of conductors for general sextics, whereas for cubics both were identical. |
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Dan (02/02/22):
now I have completely analyzed all of Megrez's
(ftp://megrez.math.u-bordeaux.fr/pub/numberfields/degree6) sextic fields N with Galois group Gal(N|Q) of order 6. Such fields occur only for the signatures (0,3), totally complex (/index60/t60.001) (6,0), totally real (/index66/t66.001) and are sawn rather thin, since the overwhelming part of the fields, also for the other signatures (2,2) and (4,1), consists of S_6 fields ( #Gal(N|Q) = 720 ). This phenomenon is similar as for M. Pohst's quintics, computed 1994/98 with the aid of KANT (probably a rival of PARI). In the following results I use the designations D ... discriminant of the sextic field N K ... quadratic subfield of N f ... conductor of N|K as an integral ideal of K d ... discriminant of K ( thus, D = f^4*d^3, by Hilbert's Theorem 39, more exactly, D = d^3*Norm_{K|Q}(f^2) ) s = sigma_3 ... modified 3-class rank of K r = rho_3 ... 3-class rank of K t'', w'' as in my communication of 2002/02/14 ( t''+w'' = dim_{F_3}( K(f) / S_f*K(f)^3 ) is the 3-rank of the ray generator class group mod f of K ) d' = delta'_3(f) = dim_{F_3}( I_3(f) / I_3(f)\cap S_f*K(f)^3 ) ... 3-ray defect mod f of K r' = rho_3(f) ... 3-ray class rank modulo f of K ( by my formula in the mentioned communication, we have r' = r + t'' + w'' - d' ) Gal = Gal(N|Q) ... absolute Galois group of N decomp ... decomposition behavior of the rational prime divisor(s) of the conductor f adm ... 3-admissibility of the rational prime divisor(s) of the conductor f rem ... remark giving the type of N as compositum or normal field N(...), where L means non-Galois cubic, L' cyclic cubic, and subscripts indicate the discriminant (1) 10 Totally Complex Fields -------------------------------------------------------------------------- D | f | d | s || r | t'' | w'' | d' | r' || Gal || decomp | adm | rem -----------------------------||-------------------------||-----||--------- -12167 | (1) | -23 | 1 || 1 | 0 | 0 | 0 | 1 || D_3 || - | yes | N(L_-23) -16807 | q_7 | -7 | 0 || 0 | 1 | 0 | 0 | 1 || C_6 || ramif. | no | K_-7*L'_49 -19683 | q_3^3 | -3 | 1 || 0 | 0 | 2 | 1 | 1 || C_6 || ramif. | yes | K_-3*L'_81 -21296 | (2) | -11 | 0 || 0 | 1 | 0 | 0 | 1 || D_3 || inert | yes | N(L_-44) -29791 | (1) | -31 | 1 || 1 | 0 | 0 | 0 | 1 || D_3 || - | yes | N(L_-31) -34992 |(2)*q_3^2 | -3 | 1 || 0 | 1 | 1 | 1 | 1 || D_3 || 2i,3r | yes | N(Q(2^1/3)) -64827 | q_7*q'_7 | -3 | 1 || 0 | 2 | 0 | 1 | 1 || C_6 || split | yes | K_-3*L'_49 -109744 | (2) | -19 | 0 || 0 | 1 | 0 | 0 | 1 || D_3 || inert | yes | N(L_-76) -153664 | (7) | -4 | 0 || 0 | 1 | 0 | 0 | 1 || C_6 || inert | no | K_-4*L'_49 -177147 | q_3^4 | -3 | 1 || 0 | 0 | 3 | 1 | 2 || D_3 || ramif. | yes | N(Q(3^1/3)) -------------------------------------------------------------------------- (2) 14 Totally Real Fields -------------------------------------------------------------------------- D | f | d | s || r | t'' | w'' | d' | r' || Gal || decomp | adm | rem -----------------------------||-------------------------||-----||--------- 300125 | (7) | 5 | 1 || 0 | 1 | 0 | 0 | 1 || C_6 || inert | no | K_5*L'_49 371293 | q_13 | 13 | 1 || 0 | 1 | 0 | 0 | 1 || C_6 || ramif. | no | K_13*L'_169 453789 | q_7 | 21 | 1 || 0 | 1 | 0 | 0 | 1 || C_6 || ramif. | no | K_21*L'_49 810448 | (2) | 37 | 1 || 0 | 1 | 0 | 0 | 1 || D_3 || inert | yes | N(L_148) 820125 | (3^2) | 5 | 1 || 0 | 0 | 2 | 1 | 1 || C_3 || inert |3n,9y| K_5*L'_81 1075648 | q_7 | 28 | 1 || 0 | 1 | 0 | 0 | 1 || C_3 || ramif. | no | K_28*L'_49 1229312 | q_7*q'_7 | 8 | 1 || 0 | 2 | 0 | 1 | 1 || C_6 || split | yes | K_8*L'_49 1259712 | q_3^3 | 12 | 1 || 0 | 0 | 2 | 1 | 1 || C_3 || ramif. | yes | K_12*L'_81 3359232 | (3^2) | 8 | 1 || 0 | 0 | 2 | 1 | 1 || C_6 || inert |3n,9y| K_8*L'_81 3570125 | (13) | 5 | 1 || 0 | 1 | 0 | 0 | 1 || C_3 || inert | no | K_5*L'_169 4148928 | (7) | 12 | 1 || 0 | 1 | 0 | 0 | 1 || C_6 || inert | no | K_12*L'_49 5274997 | (7) | 13 | 1 || 0 | 1 | 0 | 0 | 1 || C_6 || inert | no | K_13*L'_49 6751269 | q_3^3 | 21 | 1 || 0 | 0 | 2 | 1 | 1 || C_6 || ramif. | yes | K_21*L'_81 8605184 | q_7 | 56 | 1 || 0 | 1 | 0 | 0 | 1 || C_6 || ramif. | no | K_56*L'_49 -------------------------------------------------------------------------- We observe the following cases: 1. Rational integer conductors of Hilbert and ring class fields: a) f = 1 ==> D_3 normal field N (Hilbert class field): -12167, N(L_-23); -29791, N(L_-31). b) f = free 3-admissible inert prime = -1(mod 3), here always = 2 ==> D_3 normal field N (ring class field): -21296, N(L_-44); -109744, N(L_-76); 810448, N(L_148). c) f = product of restrictive 3-admissible inert and ramified primes 2,3 ==> D_3 normal field N (ring class field): -34992, N(Q(2^1/3)). d) f = power of restrictive 3-admissible ramified prime 3 ==> D_3 normal field N (ring class field): -177147, N(Q(3^1/3)). 2. Rational integer conductors of ray class fields: a) f = restrictive 3-admissible square of inert prime 3 ==> C_6 compositum N (ray class field), d' = 1 excludes a D_3 normal field: 820125, K_5*L'_81; 3359232, K_8*L'_81. b) f = restrictive 3-admissible split prime = +1(mod 3), here always = (7) = q_7*q'_7 ==> C_6 compositum N (ray class field), d' = 1 excludes a D_3 normal field: -64827, K_-3*L'_49; 1229312, K_8*L'_49. c) f = not 3-admissible inert prime = +1(mod 3), here either 7 or 13 ==> C_6 compositum N (ray class field): -153664, K_-4*L'_49; 300125, K_5*L'_49; 3570125, K_5*L'_169; 4148928, K_12*L'_49; 5274997, K_13*L'_49. 3. Quadratic ideal conductors of ray class fields: a) f = prime ideal factor of not 3-admissible ramified prime = +1(mod 3), here either 7 or 13 ==> C_6 compositum N (ray class field): -16807, K_-7*L'_49; 371293, K_13*L'_169; 453789, K_21*L'_49; 1075648, K_28*L'_49; 8605184, K_56*L'_49. b) f = power of prime ideal factor of restrictive 3-admissible ramified prime 3 ==> C_6 compositum N (ray class field): -19683, K_-3*L'_81; 1259712, K_12*L'_81; 6751269, K_21*L'_81. REMARK concerning -177147: This is the only case where r' = 2, since the conductor is irregular. We have discussed this case in our communications of 2002/02/10 and 12. Besides the field N(Q(3^1/3)) = K((j(j-1))^1/3), the 3-ray class field F' mod 9 of K = Q(-3^1/2) = Q(j) contains 2 further Kummer fields, as you communicated to me: N' = K((j^2(j-1))^1/3) and N'' = K((j-1)^1/3) ~ N'. Here, the multiplicity of the conductor 9 over d = -3 is m(-3,9) = 3, but the multiplicity of the discriminant -177147 is only m(-177147) = 2, since N'' ~ N', as you told me. |
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