1. Transfers ("Verlagerungen").Our Target.If the 3class group Syl_{3}(C(K)) of an arbitrary base field K is of type (3,3), then the structure of the automorphism group G = Gal(K_{2}K) of the 2nd Hilbert 3class field K_{2} of K over K as a 2stage metabelian 3group in ZEF determines the principalization type (k(1),...,k(4)) of K in the four unramified cyclic cubic extensions N_{1},...,N_{4} of K.
How do we approach this target? Artin's Idea. For this enterprise it is necessary to investigate the transfer V_{G,M} of the commutator factor group G/G' to the commutator factor group M/M' for each of the 4 maximal subgroups M = M_{j} (1 <= j <= 4) of G. Class field theory establishes the connection between the transfer V_{G,M} : G/G' > M/M' and the class extension homomorphism j_{NK} : Syl_{3}(C(K)) > Syl_{3}(C(N)), since G/G' ~ Gal(K_{1}K) ~ Syl_{3}(C(K)) and M_{j}/M_{j}' ~ Gal((N_{j})_{1}N_{j}) ~ Syl_{3}(C(N_{j})).
PROPOSITION 1.1. Assumptions: Let m,n be rational integers, such that 2 <= m <= n and let G be a 2stage metabelian 3group in ZEF(m,n) with generators x,y, such that the 4 maximal subgroups M_{j} (1 <= j <= 4) of G are given by M_{j} = < g(j), G' > where g(1) = x, g(2) = y, g(3) = xy, g(4) = xy^{1}. Finally, denote by V_{j} the transfer from G/G' to M_{j}/M_{j}'. Claims: 1. The commutator subgroups are M_{j}' = (G')^{g(j)1} (1 <= j <= 4). 2. The transfers are given by a) V_{j}(gG') = g^{3}M_{j}' for g in G  M_{j} b) V_{j}(uG') = u^{1+h+h2}M_{j}' for u in M_{j} with arbitrary h in G  M_{j} Remark: u^{1+h+h2} = u^{3}[u,h]^{3}[[u,h],h] (mod M_{j}'). 

2. The Class Groups of N_{1},...,N_{4}.Commutator Subgroups of the Maximal Subgroups.We can considerably refine claim 1 of the preceding Proposition 1.1. THEOREM 2.1. Assumptions: Let m be a rational integer, such that m >= 3, and let G = G^{(m)}(a,b,c) be a 2stage metabelian 3group of maximal class in ZEF(m,m) with generators s,s_{1}, where we assume s in G  C_{2} and s_{1} in C_{2}  G' in the case of m > 3, i. e., G' < C_{2} < G, such that the 4 maximal subgroups M_{j} (1 <= j <= 4) of G are given by M_{j} = < g(j), G' > where g(1) = s_{1}, g(2) = s, g(3) = ss_{1}, g(4) = ss_{1}^{1}. Claims: 1. The commutator subgroups are M_{j}' = G_{3} and thus Syl_{3}(C(N_{j})) = M_{j}/M_{j}' = (3,3) uniformly for 2 <= j <= 4. 2. An exceptional role is played by M_{1}' = < s_{m1}^{c} >: a) M_{1}' = 1 and Syl_{3}(C(N_{1})) = M_{1} = A(3,m1) (for m = 4, we can also have M_{1} = (3,3,3)) for c = 0, i. e., G in ZEF a(m,m) b) M_{1}' = G_{m1} = (3) and Syl_{3}(C(N_{1})) = M_{1}/M_{1}' = A(3,m2) for c != 0, i. e., G in ZEF b(m,m) 3. M_{1} is an abelian normal subgroup of G for c = 0. 4. The M_{j} with j != 1 are abelian normal subgroups of G iff m = 3.
THEOREM 2.2. Assumptions: Let m,n be a rational integers, such that 4 <= m < n <=2m  3, i. e., e = n  m + 2 >= 3, and let G = G^{(m,n)}((a,b,c,d),r) be a 2stage metabelian 3group of nonmaximal class in ZEF(m,n) with generators x,y, such that G_{3} = < x^{3},y^{3},G_{4} >, where we assume x in G  C_{s} and y in C_{s}  G' in the case of s < m  1, i. e., G in ZEF 2(m,n), such that the 4 maximal subgroups M_{j} (1 <= j <= 4) of G are given by M_{j} = < g(j), G' > where g(1) = y, g(2) = x, g(3) = xy, g(4) = xy^{1}. Claims: Introducing some additional notation v_{3} = [y,x]^{x1}, w_{3} = [y,x]^{y1}, s_{4} = (y^{3})^{x1}, s_{5} = s_{4}^{x1}, t_{4} = (x^{3})^{y1}, t_{5} = t_{4}^{y1}, S_{4} = < s_{4} > * < s_{5} >, T_{4} = < t_{4} > * < t_{5} > we have: 1. M_{1}' = < w_{3},T_{4} > and a) Syl_{3}(C(N_{1})) = M_{1}/M_{1}' = A(3,m1) for r = 0, i. e., G in ZEF a(m,n) (for m = 4, we can also have M_{1}/M_{1}' = (3,3,3)) b) Syl_{3}(C(N_{1})) = M_{1}/M_{1}' = A(3,m2) for r != 0, i. e., G in ZEF b(m,n) (for m = 5, we can also have M_{1}/M_{1}' = (3,3,3)) 2. M_{2}' = < v_{3},S_{4} > and Syl_{3}(C(N_{2})) = M_{2}/M_{2}' = A(3,e) (for e = 3, we can also have M_{2}/M_{2}' = (3,3,3)) 3. M_{3}' = < v_{3}w_{3},G_{4} > and Syl_{3}(C(N_{3})) = M_{3}/M_{3}' = (9,3) or (3,3,3) 4. M_{4}' = < v_{3}w_{3}^{1},G_{4} > and Syl_{3}(C(N_{4})) = M_{4}/M_{4}' = (9,3) or (3,3,3) 5. In particular, none of the maximal normal subgroups M_{1},...,M_{4} is abelian.
Insider's Know How. In the special case of a quadratic base field K, the structure of G = Gal(K_{2}K) determines also the family of 3class numbers (h_{1},...,h_{4}) of the absolute cubic subfields L_{1},...,L_{4} of the normal S_{3}fields N_{1},...,N_{4} between K_{1} and K, and, to a certain extent, also vice versa. For the proof of the following theorem we use the class number formulas of Scholz for the principal factorization types (PFT) Alpha and Delta. THEOREM 2.3. Assumptions: Let K be a quadratic number field with discriminant d and with 3class group of type (3,3). Denote by G = Gal(K_{2}K) in ZEF(m,n) with 3 <= m <= n the automorphism group of the 2^{nd} Hilbert 3class field K_{2} of K over K. Claims: 1. If G is of maximal class, i. e., m = n, then K must be a real quadratic field with d > 0 and (h_{1},...,h_{4}) = (3^{q},3,3,3), where the exponent q is given by [a) if G in ZEF 2a(m,m) with odd m, then q = (m  1)/2 and L_{1} is of PFT Alpha, a case which is conjectured to be impossible and did never occur in numerical examples,] b) if G in ZEF 2a(m,m) with even m, then q = (m  2)/2 and L_{1} is of PFT Delta, c) if G in ZEF 2b(m,m) with even m, then q = (m  2)/2 and L_{1} is of PFT Alpha, [d) if G in ZEF 2b(m,m) with odd m, then q = (m  3)/2 and L_{1} is of PFT Delta, but this case is definitely impossible, since G in ZEF 2b(m,m) implies total principalization in L_{1}.] L_{2},L_{3},L_{4} are always of PFT Alpha. 2. If G is not of maximal class, i. e., 4 <= m < n, resp. e = n  m + 2 >= 3, and if K is a complex quadratic field with d < 0, then L_{1},...,L_{4} are of PFT Alpha, e must be odd, G is either in ZEF a(m,n) with even m or in ZEF b(m,n) with odd m, and (h_{1},...,h_{4}) = (3^{q(1)},3^{q(2)},3,3), where the exponents are given by a) if G in ZEF a(m,n) with even m, then q(1) = (m  2)/2 b) if G in ZEF b(m,n) with odd m, then q(1) = (m  3)/2 c) q(2) = (e  1)/2 3. If K is a complex quadratic field with d < 0, then G is not of maximal class, i. e., 4 <= m < n, resp. e = n  m + 2 >= 3, e = 2*q(2) + 1 must be odd, G is either in ZEF a(m,n) with even m = 2*q(1) + 2 or in ZEF b(m,n) with odd m = 2*q(1) + 3, and we have the following particular implications: a) if (h_{1},...,h_{4}) = (3,3,3,3), i. e., q(1) = q(2) = 1, then G in ZEF 1a(4,5) or ZEF 1b(5,6) b) if (h_{1},...,h_{4}) = (9,3,3,3), i. e., q(1) = 2, q(2) = 1, then G in ZEF 2a(6,7) or ZEF 2b(7,8) c) if (h_{1},...,h_{4}) = (9,9,3,3), i. e., q(1) = q(2) = 2, then G in ZEF 1a(6,9) or ZEF 1b(7,10) d) if (h_{1},...,h_{4}) = (27,9,3,3), i. e., q(1) = 3, q(2) = 2, then G in ZEF 2a(8,11) or ZEF 2b(9,12) 

3. Principalization Types.Explicit terms for the images under the transfers.PROPOSITION 3.1. Assumptions: Let m >= 3 be a rational integer and G = G^{(m)}(a,b,c) in ZEF (m,m) a 2stage metabelian 3group of maximal class with generators s,s_{1}, where we assume s in G  C_{2} and s_{1} in C_{2}  G' in the case of m > 3, i. e., G' < C_{2} < G. Finally, let g be an element of G with representation g = s^{i}s_{1}^{k} (mod G'), where 1 <= i,k <= 1, and denote by V_{j} the transfer from G/G' to M_{j}/M_{j}' (1 <= j <= 4). Claims: 1. V_{1}(g*G') = s_{m1}^{ai+bk}*< s_{m1}^{c} > 2. a) V_{j}(g*G') = s_{2}^{ai+bk}*1 for 2 <= j <= 4, if m = 3 b) V_{j}(g*G') = 1*G_{3} for 2 <= j <= 4, if m >= 4 The transfer kernels. THEOREM 3.2. Assumptions: similar as in Proposition 3.1. Claims: 1. Cases having the total preimage group G/G' as kernel: a) c != 0 ==> Ker V_{1} = G/G' b) a = b = 0 ==> Ker V_{1} = G/G' c) a = b = 0, m = 3 ==> Ker V_{j} = G/G' for 1 <= j <= 4 d) m >= 4 ==> Ker V_{j} = G/G' for 2 <= j <= 4 2. Cases having a nontrivial subgroup of G/G' as kernel: a) (a,b,c) = (1,0,0) ==> Ker V_{1} = M_{1}/G' b) (a,b,c) = (1,0,0), m = 3 ==> Ker V_{j} = M_{1}/G' for 1 <= j <= 4 c) (a,b,c) = (0,+1,0) ==> Ker V_{1} = M_{2}/G' 3. The Principalization Types for Groups of Maximal Class are: a) (a,b,c) = (1,0,0), m = 3 ==> (k(1),...,k(4)) = (1,1,1,1) b) (a,b,c) = (1,0,0), m >= 4 ==> (k(1),...,k(4)) = (1,0,0,0) c) (a,b,c) = (0,+1,0), m >= 4 ==> (k(1),...,k(4)) = (0,1,0,0) d) (a,b,c) = (0,0,0) ==> (k(1),...,k(4)) = (0,0,0,0) e) c != 0 ==> (k(1),...,k(4)) = (0,0,0,0) 


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