 # p-Ray Class Fields

## Synthesis and Enumeration

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 p-ray 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 p-class rank r = 0and let f be the conductor of a cyclic field L'of degree p and of multiplicity 1,i. e., either f = p2 or a rational prime f = q = 1 (mod p).

Then the p-ray 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) = (zeta-1)2 ramifies in K, then F' contains
a D(6)-field N with discriminant d(N) = -27*94 and relative conductor 9,
the compositum N' = K*L' with discriminant d(N') = -27*272 and relative conductor 3*(zeta-1),
and 2 isomorphic non-Galois fields N(i) (i=1,2) with discriminant d(N(i)) = -27*94 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*q4 and relative conductor q,
the compositum N' = K*L' with discriminant d(N') = -27*q4 and relative conductor q,
and 2 isomorphic non-Galois fields N(i) (i=1,2) with discriminant d(N(i)) = -27*q2 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*q4 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) = dp*q2(p-1) and relative conductor q,
the compositum N' = K*L' with discriminant d(N') = dp*q2(p-1) and relative conductor q,
2 isomorphic non-Galois fields N(i) (i=1,2) with discriminant d(N(i)) = dp*qp-1 and relative conductors Q resp. Q',
and p-3 non-Galois fields N'(i) (i=1,...,p-3) with discriminant d(N'(i)) = dp*q2(p-1) and relative conductor q;

 F'(Q*Q') / / \ \ \ \ \ F'(Q) F'(Q') F(q) | | | | | | | | | | | N(1) N(2) N N' N'(1) ... N'(p-3) \ \ / | / | / / / K L L' | / / Q

b) if f = q = 1 (mod p) and (q) = Q2 ramifies in K, then F' contains only
the compositum N' = K*L' with discriminant d(N') = dp*qp-1 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') = dp*q2(p-1) and relative conductor q;

d) if f = p2 and (p) = P*P' splits in K, then F' = F'(P2*P'2) contains
a D(2p)-field N with discriminant d(N) = dp*p4(p-1) and relative conductor p2,
the compositum N' = K*L' with discriminant d(N') = dp*p4(p-1) and relative conductor p2,
2 isomorphic non-Galois fields N(i) (i=1,2) with discriminant d(N(i)) = dp*p2(p-1) and relative conductors P2 resp. P'2,
and p-3 non-Galois fields N'(i) (i=1,...,p-3) with discriminant d(N'(i)) = dp*p4(p-1) and relative conductor p2.

 F'(P2*P'2) / / \ \ \ \ \ F'(P2) F'(P'2) F(p2) | | | | | | | | | | | N(1) N(2) N N' N'(1) ... N'(p-3) \ \ / | / | / / / K L L' | / / Q

e) if f = p2 and (p) = P remains inert in K, then F' contains
a D(2p)-field N with discriminant d(N) = dp*p4(p-1) and relative conductor p2,
the compositum N' = K*L' with discriminant d(N') = dp*p4(p-1) and relative conductor p2,
and p-1 non-Galois fields N(i) (i=1,...,p-1) with discriminant d(N(i)) = dp*p4(p-1) and relative conductor p2.

f) if f = p2 and (p) = P2 ramifies in K, then F' = F'(P4) contains
a D(2p)-field N with discriminant d(N) = dp*p2(p-1) and relative conductor p,
the compositum N' = K*L' with discriminant d(N') = dp*p3(p-1) and relative conductor P3,
and p-1 non-Galois fields N(i) (i=1,...,p-1) with discriminant d(N(i)) = dp*p3(p-1) and relative conductor P3.
In the irregular case d = -3 (mod 9) and p = 3, F' = F'(P4) contains additionally
p D(2p)-fields N~(i) (i=1,...,p) with discriminant d(N~(i)) = dp*p4(p-1) and relative conductor p2,
and p*(p-1) non-Galois fields N'(i) (i=1,...,p*(p-1)) with discriminant d(N'(i)) = dp*p4(p-1) and relative conductor p2.

and p*(p-1) non-Galois fields N'(i) (i=1,...,p*(p-1)) with discriminant d(N'(i)) = dp*p4(p-1) and relative conductor p2.

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