1. Unramified Cyclic Cubic Extensions K_{6} of a Quadratic Field K_{2}.A quadratic field K_{2} has no, resp. one, resp. more than oneunramified cyclic cubic extension K_{6}, iff the 3class rank of K_{2} equals 0, resp. 1, resp. exceeds 1. Since an unramified extension has relative conductor f = 1 without any prime divisors, the only possibility for ambiguous principal ideals of K_{6} are extension ideals of K_{2} that become principal in K_{6}. This fact excludes the Principal Factorization Types BETA, GAMMA, EPSILON, discussed in the previous section and only the Types ALPHA and DELTA remain. Thus we have three possible cases of class number relations: 1. Either a real quadratic type ALPHA field with D_{2} > 0 and h_{6} = (1 / 9)*h_{2}*h_{3}^{2} 2. Or a real quadratic type DELTA field with D_{2} > 0 and h_{6} = (1 / 3)*h_{2}*h_{3}^{2} 3. Or a complex quadratic type ALPHA field with D_{2} < 0 and h_{6} = (1 / 3)*h_{2}*h_{3}^{2} 

2. The Galois Group of (K_{6})_{1} over K_{2}.Arnold Scholz investigated the Galois group G = Gal((K_{6})_{1}K_{2}) ofthe 1^{st} Hilbert 3class field (K_{6})_{1} of K_{6} over K_{2} to get the structure of its abelian normal subgroup A = Gal((K_{6})_{1}K_{6}) of index 3, which is isomorphic to the 3class group Syl_{3}C(K_{6}) of K_{6}.
This is an application of class field theory, since (K_{2})_{1} is the maximal abelian unramified 3extension of K_{2} and thus the subgroup U = Gal((K_{6})_{1}(K_{2})_{1}) of G = Gal((K_{6})_{1}K_{2}) with factor group G/U = Gal((K_{2})_{1}K_{2}) = Syl_{3}C(K_{2}) must be the minimal subgroup of G with abelian factor group, i. e., must coincide with the commutator subgroup G' of G. Further, G is a 2stage metabelian 3group, since G' = Gal((K_{6})_{1}(K_{2})_{1}) as a subgroup of the abelian normal subgroup A = Gal((K_{6})_{1}K_{6}) = Syl_{3}C(K_{6}) is abelian, i.e., G'' = 1. 2.1. Quadratic Fields K_{2} of 3Class Rank 1.First, Scholz shortly indicates, that the case whereK_{2} is a quadratic field of 3class rank 1, i. e., with 3class group of type (3^{x}) (x >= 1) and thus with 3class number h_{2} = 3^{x} is trivial: by assumption, we have G/G' = (3^{x}) and since in the present context the Frattini subgroup has the shape F(G) = G'*G^{3}, we get the 3elementary factor group G/F(G) = G/G'*G^{3} = (3) of 3rank 1. According to the Basis Theorem of Burnside, the minimal number of generators of G is 1, i. e., G is cyclic, G' = 1, (K_{6})_{1} = (K_{2})_{1}, G = (3^{x}), Syl_{3}C(K_{6}) = A = (3^{x1}), and by the class number relation (note that type ALPHA is impossible here for D_{2} > 0) 3^{x1} = h_{6} = (1/3)*h_{2}*h_{3}^{2} = (1/3)*3^{x}*h_{3}^{2} and finally h_{3} = 1.
2.2. Quadratic Fields K_{2} of 3Class Rank 2.Next, to be nontrivial but specific, Scholz assumes that K_{2} is a quadratic fieldwith 3class group of (elementary abelian) type (3,3), and therefore h_{2} = 9. For this situation, we can determine the structure of the Galois group G = Gal( (K_{6})_{1}  K_{2} ) of the 1st Hilbert 3class field (K_{6})_{1} of K_{6} over K_{2}. All we need to know is (1) the 3class number h_{3} of the absolute cubic subfield K_{3} of K_{6} (2) the single principalization type k(j) of K_{2} in K_{6} only, i. e., only the relevant member of the full quadruplet principalization type (k(1),...,k(4)) of K_{2}. In all numerical examples that occurred up to now in the real and complex case, only the following few configurations were discovered. 1. h_{3} = 3 and k(j) != 0 is not a fixed point (k(j) != j) then C_{6} = (9,3) and G = G^{(4)}(0,+1,0) in ZEF 2a(4,4) < CF(4,4,3) 2. h_{3} = 3 and k(j) != 0 is a fixed point (k(j) = j) then C_{6} = (9,3) and G = G^{(4)}(1,0,0) in ZEF 2a(4,4) < CF(4,4,3) 3. h_{3} = 3 and k(j) = 0 then C_{6} = (3,3) and G = G^{(3)}(0,0,0) in ZEF 1a(3,3) < CF(3,3,3)
4. h_{3} = 9 and k(j) != 0 is not a fixed point (k(j) != j) then C_{6} = (27,9) and G = G^{(6)}(0,+1,0) in ZEF 2a(6,6) < CF(6,6,3) 5. h_{3} = 9 and k(j) != 0 is a fixed point (k(j) = j) then C_{6} = (27,9) and G = G^{(6)}(1,0,0) in ZEF 2a(6,6) < CF(6,6,3) 6. h_{3} = 9 and k(j) = 0 then C_{6} = (9,9) and G = G^{(5)}(0,0,0) in ZEF 2a(5,5) < CF(5,5,3) 7. h_{3} = 27 and k(j) != 0 is not a fixed point (k(j) != j) then C_{6} = (81,27) and G = G^{(8)}(0,+1,0) in ZEF 2a(8,8) < CF(8,8,3)
Remarks: 1) The 3class group C_{3} of the absolute cubic subfield is always cyclic. 2) Group theoretically, there is a difference between the totally real principalization types (2,0,0,0), i. e. a.1, and (1,0,0,0), i. e. a.2. The three sextic fields with k(j) = 0 always belong to configuration 3, but the exceptional field with k(j) != 0 to 1 in the former case, and to 2 in the latter. 3) For the totally real principalization type (0,0,0,0), i. e. a.3, three sextic fields belong to configuration 3, and one exceptional field belongs to 6. 4) The most frequent complex principalization type is D, where all four sextic fields belong to configuration 1, except the fixed points, which belong to 2. 5) For the complex principalization types H and G.19, all four sextic fields belong to configuration 1. 6) Configurations 4, resp. 5, occur for certain complex principalization types: a single exceptional field for principalization types E and G.16 and the first variant of type H, and two exceptional fields for principalization type F, the second variant of type H, and the variants of types G.16 and G.19. 7) Finally, configuration 7 is unique, up to now. It occurs for a single field with discriminant D_{3} = 159208 and with a variant of principalization type F.13. 


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