 # Theorems concerning the impact of 3-class numbers.

 Remark. For the statement of the theorems we must recall some concepts. An unramified cyclic cubic relative extension N of a real quadratic field K with relative automorphism group G = Gal(N|K) and unit group UN is called of principal factorization type (or simply of type) Alpha, if the cohomology group H0(G,UN) is non-trivial (i. e., if the unit norm index (UK : NormN|KUN) equals 3) and it is called of principal factorization type (or simply of type) Delta, if the cohomology group H0(G,UN) is trivial (i. e., if the unit norm index (UK : NormN|KUN) equals 1 and thus the fundamental unit of K is norm of a unit of N). * Theorem 1. (for the proof see ) Let K be a real quadratic field with 3-class group of type (3,3). Suppose that at least three of the four unramified cyclic cubic extensions N1,...,N4 of K are of type Alpha, say N2,N3,N4. Let the 3-class numbers of the non-Galois cubic subfields be (h(L1),h(L2),h(L3),h(L4)) = (3u,3,3,3) with u ≥ 1. Then the metabelian 3-group Gal(K2|K) of automorphisms of the second Hilbert 3-class field K2 of K is of maximal class and belongs to CF2a(2u+2) with symbolic order Y2u, if N1 is of type Delta, and to CF2b(2u+2) with symbolic order Y2u' and u ≥ 2, if N1 is of type Alpha. The third possibility that Gal(K2|K) belongs to CFa(2u+1) with symbolic order Y2u-1, if N1 is of type Alpha, does obviously not occur (weak leaf conjecture). * Theorem 2. (for the proof see ) Let K be either a real quadratic field with 3-class group of type (3,3) such that all four unramified cyclic cubic extensions N1,...,N4 of K are of type Delta, or a complex quadratic field with 3-class group of type (3,3). Let the 3-class numbers of the non-Galois cubic subfields be (h(L1),h(L2),h(L3),h(L4)) = (3u,3v,3,3) with u ≥ v ≥ 1. Then the metabelian 3-group Gal(K2|K) of automorphisms of the second Hilbert 3-class field K2 of K is at most of second maximal class and belongs either to CBFa(2u+2,2u+2v+1) with symbolic order R2u,2v or to CBFb(2u+3,2u+2v+2). The isomorphism invariant e has the odd value e = 2v+1.

 References:  Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, (Latest Update) Univ. Graz, 2008.

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