The following fundamental facts concerning the exact enumeration of nodes in the Abelian network over quadratic number fields K have not appeared in printed form yet. Here, they are communicated for the first time by me as a main theorem. Throughout the sequel, let p denote an odd rational prime.The following central result gives counts of all pray class fieldsover the simplest quadratics K for the simplest conductors f.It also settles the problem of degeneration when conductors are lifted. MAIN THEOREM. (Simultaneous rank increments and conductor liftings.) (D. C. Mayer, 2002/12/10) Let K be an imaginary quadratic field of pclass rank r = 0and let f be the conductor of a cyclic field L'of degree p and of multiplicity 1,i. e., either f = p^{2} or a rational prime f = q = 1 (mod p). Then the pray class field F' mod f of K contains the followingcyclic relative extensions of K of degree pwith a divisor of f as their conductor. 1. In the exceptional case d = 3 and p = 3, a) if f = 9, i. e., (3) = (zeta1)^{2} ramifies in K, then F' contains a D(6)field N with discriminant d(N) = 27*9^{4} and relative conductor 9, the compositum N' = K*L' with discriminant d(N') = 27*27^{2} and relative conductor 3*(zeta1), and 2 isomorphic nonGalois fields N(i) (i=1,2) with discriminant d(N(i)) = 27*9^{4} and relative conductor 9; b) if f = q = 1 (mod 9), i. e., (q) = Q*Q' splits in K, then F' contains a D(6)field N with discriminant d(N) = 27*q^{4} and relative conductor q, the compositum N' = K*L' with discriminant d(N') = 27*q^{4} and relative conductor q, and 2 isomorphic nonGalois fields N(i) (i=1,2) with discriminant d(N(i)) = 27*q^{2} and relative conductors Q resp. Q'; c) if f = q = 4,7 (mod 9), then F' contains only the compositum N' = K*L' with discriminant d(N') = 27*q^{4} and relative conductor q. 2. In the general case d < 3 or d = 3 and p >= 5, a) if f = q = 1 (mod p) and (q) = Q*Q' splits in K, then F' = F'(Q*Q') contains a D(2p)field N with discriminant d(N) = d^{p}*q^{2(p1)} and relative conductor q, the compositum N' = K*L' with discriminant d(N') = d^{p}*q^{2(p1)} and relative conductor q, 2 isomorphic nonGalois fields N(i) (i=1,2) with discriminant d(N(i)) = d^{p}*q^{p1} and relative conductors Q resp. Q', and p3 nonGalois fields N'(i) (i=1,...,p3) with discriminant d(N'(i)) = d^{p}*q^{2(p1)} and relative conductor q;
    F'(Q*Q')       /   /   \   \   \   \   \   F'(Q)   F'(Q')     F(q)                                    N(1)   N(2)     N   N'   N'(1)   ...   N'(p3)   \   \   /    /    /   /   /       K   L   L'        /   /       Q      b) if f = q = 1 (mod p) and (q) = Q^{2} ramifies in K, then F' contains only the compositum N' = K*L' with discriminant d(N') = d^{p}*q^{p1} and relative conductor Q; c) if f = q = 1 (mod p) and (q) = Q remains inert in K, then F' contains only the compositum N' = K*L' with discriminant d(N') = d^{p}*q^{2(p1)} and relative conductor q; d) if f = p^{2} and (p) = P*P' splits in K, then F' = F'(P^{2}*P'^{2}) contains a D(2p)field N with discriminant d(N) = d^{p}*p^{4(p1)} and relative conductor p^{2}, the compositum N' = K*L' with discriminant d(N') = d^{p}*p^{4(p1)} and relative conductor p^{2}, 2 isomorphic nonGalois fields N(i) (i=1,2) with discriminant d(N(i)) = d^{p}*p^{2(p1)} and relative conductors P^{2} resp. P'^{2}, and p3 nonGalois fields N'(i) (i=1,...,p3) with discriminant d(N'(i)) = d^{p}*p^{4(p1)} and relative conductor p^{2}.
    F'(P^{2}*P'^{2})       /   /   \   \   \   \   \   F'(P^{2})   F'(P'^{2})     F(p^{2})                                    N(1)   N(2)     N   N'   N'(1)   ...   N'(p3)   \   \   /    /    /   /   /       K   L   L'        /   /       Q      e) if f = p^{2} and (p) = P remains inert in K, then F' contains a D(2p)field N with discriminant d(N) = d^{p}*p^{4(p1)} and relative conductor p^{2}, the compositum N' = K*L' with discriminant d(N') = d^{p}*p^{4(p1)} and relative conductor p^{2}, and p1 nonGalois fields N(i) (i=1,...,p1) with discriminant d(N(i)) = d^{p}*p^{4(p1)} and relative conductor p^{2}. f) if f = p^{2} and (p) = P^{2} ramifies in K, then F' = F'(P^{4}) contains a D(2p)field N with discriminant d(N) = d^{p}*p^{2(p1)} and relative conductor p, the compositum N' = K*L' with discriminant d(N') = d^{p}*p^{3(p1)} and relative conductor P^{3}, and p1 nonGalois fields N(i) (i=1,...,p1) with discriminant d(N(i)) = d^{p}*p^{3(p1)} and relative conductor P^{3}. In the irregular case d = 3 (mod 9) and p = 3, F' = F'(P^{4}) contains additionally p D(2p)fields N~(i) (i=1,...,p) with discriminant d(N~(i)) = d^{p}*p^{4(p1)} and relative conductor p^{2}, and p*(p1) nonGalois fields N'(i) (i=1,...,p*(p1)) with discriminant d(N'(i)) = d^{p}*p^{4(p1)} and relative conductor p^{2}. and p*(p1) nonGalois fields N'(i) (i=1,...,p*(p1)) with discriminant d(N'(i)) = d^{p}*p^{4(p1)} and relative conductor p^{2}.
