1. Galois Correspondence.Arnold Scholz uses the following notation [0]for S_{3}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 T^{2} = E, S^{3} = E, TS = S^{2}T, where E denotes the identity automorphism. The following diagram summarizes the Galois correspondence:
The discriminants are connected by the relations D_{3} = f^{2}D_{2} (Hasse) and D_{6} = D_{2}^{3}(f^{2})^{2} (Hilbert) = D_{2}D_{3}^{2}, whence K_{6} is a 3ring class field with conductor f over the quadratic base field K_{2}. 

2. Expansion of Old Units.According to Dirichlet ,the torsion free unit groups of K_{2}, K_{3}, K_{6} are of the following shape, depending on the signatures: in the complex case, D_{2} < 0, U_{2} = < 1 >, U_{3} = < e_{3} >, U_{6} = < e_{6}, e_{6}^{S} > and in the totally real case, D_{2} > 0, U_{2} = < e >, U_{3} = < e_{1}, e_{2} >, U_{6} = < e, eta_{1}, eta_{2}, eta_{1}^{S}, eta_{2}^{S} > In general, the units of K_{6} cannot be completely generated by the units of all subfields and their conjugates, which are called old units by Arnold Scholz: U_{o} = U_{2}*U_{3}*U_{3}^{S}, and in particular U_{o} = < e_{3}, e_{3}^{S} > for D_{2} < 0, and U_{o} = < e, e_{1}, e_{2}, e_{1}^{S}, e_{2}^{S} > for D_{2} > 0. u = (U_{6}:U_{o}) is called the index of old units and may take the values 1,3 for D_{2} < 0, and 1,3,9 for D_{2} > 0. The regulators of U_{2}, U_{3}, U_{o}, U_{6} are connected by the following relations: R_{o} = 3*R_{2}R_{3}^{2} with R_{2} = 1 for D_{2} < 0, and R_{o} = 9*R_{2}R_{3}^{2} for D_{2} > 0 is the volume of the coarse lattice mesh of the old unit group U_{o} in logarithmic space, and for both signatures we have R_{6} = (1 / u)*R_{o} as the volume of the finer lattice mesh of the full unit group U_{6} in logarithmic space. Of course, the smaller regulator R_{6} <= R_{o} corresponds to an expansion of the old units U_{6} >= U_{o}. 

3. Contraction of Ideal Class Groups.Using the analytical class number formula,Arnold Scholz derived the following class number relations from the regulator formulas: h_{6} = (u / 3)*h_{2}h_{3}^{2} for D_{2} < 0, and h_{6} = (u / 9)*h_{2}h_{3}^{2} for D_{2} > 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 K_{6}K_{2}, which are simply coarse variants of the principal factorization types: In the complex case, we have 2 types: Type ALPHA <==> u = 1 <==> e_{3} = e_{6} <==> h_{6} = (1 / 3)*h_{2}h_{3}^{2} Type BETA/GAMMA <==> u = 3 <==> e_{3} = e_{6}^{S1} <==> h_{6} = h_{2}h_{3}^{2} For Type BETA, resp. GAMMA, Norm_{62}(e_{6}) = e_{6}^{1+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 <==> e_{1} = eta_{1} and e_{2} = eta_{2} <==> h_{6} = (1 / 9)*h_{2}h_{3}^{2} Type BETA/DELTA <==> u = 3 <==> e_{1} = eta_{1} but e_{2} = eta_{2}^{S1} <==> h_{6} = (1 / 3)*h_{2}h_{3}^{2} Type GAMMA/EPSILON <==> u = 9 <==> e_{1} = eta_{1}^{S1} and e_{2} = eta_{2}^{S1} <==> h_{6} = h_{2}h_{3}^{2} For Type ALPHA/BETA/GAMMA, resp. DELTA/EPSILON, Norm_{62}(eta_{2}) = eta_{2}^{1+S+S2} = 1, resp. e, the fundamental unit of K_{2}, whereas always Norm_{62}(eta_{1}) = eta_{1}^{1+S+S2} = 1. Arnold Scholz gives an interpretation of these results: wheras for all primes p != 3, the pclass group of K_{6} is a direct product C_{6} = C_{2}*C_{3}*C_{3}^{S}, we have a contraction of the 3class groups for the complex type ALPHA end the totally real types ALPHA, BETA, DELTA, since certainly C_{2} is contained in C_{3}*C_{3}^{S} and possibly C_{3} and C_{3}^{S} have a nontrivial intersection. 

4. The Galois Group of (K_{6})_{1} over K_{2}.This will be the topic of our next article. 


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