# p-Ray Class Fields

The internet as a presentation medium permits a delay free, almost instantaneous publication of current, brand new, and most recent scientific discoveries. Free of all the effects of retardation, delay, and backlog connected with the process of publishing in journals.

Revolutionary Results.

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 fields over 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 = 0 and 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 following cyclic relative extensions of K of degree p with 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.

 <| Navigation Center <| <| Back to Algebra <| <| Back to Curriculum <|