1. Galois Correspondence.Arnold Scholz uses the following notation [0]for S3-fields and their automorphism groups: Indices of subfields denote their absolute degrees over the rational number field P (capital Rho instead of Q). The generating automorphisms satisfy the relations T2 = E, S3 = E, TS = S2T, where E denotes the identity automorphism. The following diagram summarizes the Galois correspondence:
The discriminants are connected by the relations D3 = f2D2 (Hasse) and D6 = D23(f2)2 (Hilbert) = D2D32, whence K6 is a 3-ring class field with conductor f over the quadratic base field K2. |
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2. Expansion of Old Units.According to Dirichlet ,the torsion free unit groups of K2, K3, K6 are of the following shape, depending on the signatures: in the complex case, D2 < 0, U2 = < 1 >, U3 = < e3 >, U6 = < e6, e6S > and in the totally real case, D2 > 0, U2 = < e >, U3 = < e1, e2 >, U6 = < e, eta1, eta2, eta1S, eta2S > In general, the units of K6 cannot be completely generated by the units of all subfields and their conjugates, which are called old units by Arnold Scholz: Uo = U2*U3*U3S, and in particular Uo = < e3, e3S > for D2 < 0, and Uo = < e, e1, e2, e1S, e2S > for D2 > 0. u = (U6:Uo) is called the index of old units and may take the values 1,3 for D2 < 0, and 1,3,9 for D2 > 0. The regulators of U2, U3, Uo, U6 are connected by the following relations: Ro = 3*R2R32 with R2 = 1 for D2 < 0, and Ro = 9*R2R32 for D2 > 0 is the volume of the coarse lattice mesh of the old unit group Uo in logarithmic space, and for both signatures we have R6 = (1 / u)*Ro as the volume of the finer lattice mesh of the full unit group U6 in logarithmic space. Of course, the smaller regulator R6 <= Ro corresponds to an expansion of the old units U6 >= Uo. |
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3. Contraction of Ideal Class Groups.Using the analytical class number formula,Arnold Scholz derived the following class number relations from the regulator formulas: h6 = (u / 3)*h2h32 for D2 < 0, and h6 = (u / 9)*h2h32 for D2 > 0. According to the index u of old units and the representation of units with Hilbert's theorem 90, Scholz distinguishes several types of cyclic cubic relative extensions K6|K2, which are simply coarse variants of the principal factorization types: In the complex case, we have 2 types: Type ALPHA <==> u = 1 <==> e3 = e6 <==> h6 = (1 / 3)*h2h32 Type BETA/GAMMA <==> u = 3 <==> e3 = e6S-1 <==> h6 = h2h32 For Type BETA, resp. GAMMA, Norm6|2(e6) = e61+S+S2 = 1, resp. zeta, a primitive third root of unity. And in the totally real case, we have 3 types: Type ALPHA <==> u = 1 <==> e1 = eta1 and e2 = eta2 <==> h6 = (1 / 9)*h2h32 Type BETA/DELTA <==> u = 3 <==> e1 = eta1 but e2 = eta2S-1 <==> h6 = (1 / 3)*h2h32 Type GAMMA/EPSILON <==> u = 9 <==> e1 = eta1S-1 and e2 = eta2S-1 <==> h6 = h2h32 For Type ALPHA/BETA/GAMMA, resp. DELTA/EPSILON, Norm6|2(eta2) = eta21+S+S2 = 1, resp. e, the fundamental unit of K2, whereas always Norm6|2(eta1) = eta11+S+S2 = 1. Arnold Scholz gives an interpretation of these results: wheras for all primes p != 3, the p-class group of K6 is a direct product C6 = C2*C3*C3S, we have a contraction of the 3-class groups for the complex type ALPHA end the totally real types ALPHA, BETA, DELTA, since certainly C2 is contained in C3*C3S and possibly C3 and C3S have a non-trivial intersection. |
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4. The Galois Group of (K6)1 over K2.This will be the topic of our next article. |
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